The Algebraic and Geometric Theory of Quadratic Forms

Richard Elman

Nikita Karpenko

Alexander Merkurjev

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: rse@math.ucla.edu

Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE

E-mail address: karpenko@math.jussieu.fr

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: merkurev@math.ucla.edu

To Caroline, Tatiana and Olga

Contents

Introduction 1

Part 1. Classical theory of symmetric bilinear forms and quadratic forms 9

Chapter I. Bilinear Forms 11 1. Foundations 11 2. The Witt and Witt-Grothendieck rings of symmetric bilinear forms 19 3. Chain equivalence 21 4. Structure of the Witt ring 22 5. The Stiefel-Whitney map 28 6. Bilinear Pfister forms 32

Chapter II. Quadratic Forms 39 7. Foundations 39 8. Witt’s Theorems 46 9. Quadratic Pfister forms I 52 10. Totally singular forms 55 11. The Clifford algebra 57 12. Binary quadratic forms and quadratic algebras 60 13. The discriminant 61 14. The Clifford invariant 63 15. Chain p-equivalence of quadratic Pfister forms 64 16. Cohomological invariants 67

Chapter III. Forms over Rational Function Fields 71 17. The Cassels-Pfister Theorem 71 18. Values of forms 75 19. Forms over a discrete valuation ring 79 20. Similarities of forms 82 21. An exact sequence for W (F (t)) 88

Chapter IV. Function Fields of Quadrics 93 22. Quadrics 93 23. Quadratic Pfister forms II 98 24. Linkage of quadratic forms 101 25. The submodule J_{n}(F) 103 26. The Separation Theorem 107 27. A further characterization of quadratic Pfister forms 109 28. Excellent quadratic forms 111 29. Excellent field extensions 113 30. Central simple algebras over function fields of quadratic forms 116

Chapter V. Bilinear and Quadratic Forms and Algebraic Extensions 121 31. Structure of the Witt ring 121 32. Addendum on torsion 131 33. The total signature 133 34. Bilinear and quadratic forms under quadratic extensions 138 35. Torsion in I^{n}(F) and torsion Pfister forms 147

Chapter VI. u-invariants 161 36. The ū-invariant 161 37. The u-invariant for formally real fields 165 38. Construction of fields with even u-invariant 170 39. Addendum: Linked fields and the Hasse number 172

Chapter VII. Applications of the Milnor Conjecture 177 40. Exact sequences for quadratic extensions 177 41. Annihilators of Pfister forms 181 42. Presentation of I^{n}(F) 184 43. Going down and torsion-freeness 188

Chapter VIII. On the Norm Residue Homomorphism of Degree Two 193 44. The main theorem 193 45. Geometry of conic curves 194 46. Key exact sequence 198 47. Hilbert Theorem 90 for K_{2} 208 48. Proof of the main theorem 211

Part 2. Algebraic cycles 215

Chapter IX. Homology and Cohomology 217 49. The complex C_{∗}(X) 217 50. External products 232 51. Deformation homomorphisms 235 52. K-homology groups 238 53. Euler classes and projective bundle theorem 243 54. Chern classes 247 55. Gysin and pull-back homomorphisms 250 56. K-cohomology ring of smooth schemes 257

Chapter X. Chow Groups 261 57. Definition of Chow groups 261 58. Segre and Chern classes 268

Chapter XI. Steenrod Operations 277 59. Definition of the Steenrod operations 278 60. Properties of the Steenrod operations 281 61. Steenrod operations for smooth schemes 283

Chapter XII. Category of Chow Motives 291 62. Correspondences 291 63. Categories of correspondences 295 64. Category of Chow motives 298 65. Duality 299 66. Motives of cellular schemes 300 67. Nilpotence Theorem 302

Part 3. Quadratic forms and algebraic cycles 305

Chapter XIII. Cycles on Powers of Quadrics 307 68. Split quadrics 307 69. Isomorphisms of quadrics 309 70. Isotropic quadrics 310 71. The Chow group of dimension 0 cycles on quadrics 311 72. The reduced Chow group 313 73. Cycles on X^{2} 316

Chapter XIV. The Izhboldin Dimension 325 74. The first Witt index of subforms 325 75. Correspondences 326 76. The main theorem 329 77. Addendum: The Pythagoras number 332

Chapter XV. Application of Steenrod Operations 335 78. Computation of Steenrod operations 335 79. Values of the first Witt index 336 80. Rost correspondences 339 81. On the 2-adic order of higher Witt indices, I 342 82. Holes in I^{n} 347 83. On the 2-adic order of higher Witt indices, II 350 84. Minimal height 351

Chapter XVI. The Variety of Maximal Totally Isotropic Subspaces 355 85. The variety Gr(φ) 355 86. The Chow ring of Gr(φ) in the split case 356 87. The Chow ring of Gr(φ) in the general case 361 88. The invariant J(φ) 364 89. Steenrod operations on Ch(Gr(φ)) 366 90. Canonical dimension 367

Chapter XVII. Motives of Quadrics 371 91. Comparison of some discrete invariants of quadratic forms 371 92. The Nilpotence Theorem for quadrics 373 93. Criterion of isomorphism 375 94. Indecomposable summands 378

Appendices 381 95. Formally real fields 383 96. The space of orderings 384 97. C_{n}-fields 385 98. Algebras 387 99. Galois cohomology 393 100. Milnor K-theory of fields 397 101. The cohomology groups H^{n,i}(F, ℤ/mℤ) 402 102. Length and Herbrand index 407 103. Places 408 104. Cones and vector bundles 409 105. Group actions on algebraic schemes 418

Bibliography 421

Notation 427

Terminology 431

Introduction

The algebraic theory of quadratic forms, i.e., the study of quadratic forms over arbitrary fields, really began with the pioneering work of Witt. In his paper [139], Witt considered the totality of nondegenerate symmetric bilinear forms over an arbitrary field F of characteristic different from 2. Under this assumption, the theory of symmetric bilinear forms and the theory of quadratic forms are essentially the same. His work allowed him to form a ring W (F), now called the Witt ring, arising from the isometry classes of such forms. This work set the stage for further study. From the viewpoint of ring theory, Witt gave a presentation of this ring as a quotient of the integral group ring where the group consists of the nonzero square classes of the field F. Three methods of study arise: ring theoretic, field theoretic, i.e., the relationship of W (F) and W (K) where K is a field extension of F, and algebraic geometric. In this book, we will develop all three methods. Historically, the powerful approach using algebraic geometry has been the last to be developed. This volume attempts to show its usefulness.

The theory of quadratic forms lay dormant until the work of Cassels and then of Pfister in the 1960’s when it was still under the assumption of the field being of characteristic different from 2. Pfister employed the first two methods, ring theoretic and field theoretic, as well as a nascent algebraic geometric approach. In his postdoctoral thesis [110] Pfister determined many properties of the Witt ring. His study bifurcated into two cases: formally real fields, i.e., fields in which −1 is not a sum of squares and nonformally real fields. In particular, the Krull dimension of the Witt ring is one in the formally real case and zero otherwise. This makes the study of the interaction of bilinear forms and orderings an imperative, hence the importance of looking at real closures of the base field resulting in extensions of Sylvester’s work and Artin-Schreier theory. Pfister determined the radical, zero-divisors, and spectrum of the Witt ring. Even earlier, in [108], he discovered remarkable forms, now called Pfister forms. These are forms that are tensor products of binary forms that represent one. Pfister showed that scalar multiples of these were precisely the forms that become hyperbolic over their function field. In addition, the nonzero value set of a Pfister form is a group and in fact the group of similarity factors of the form. As an example, this applies to the quadratic form that is a sum of 2^{n} squares. Pfister also used it to show that in a nonformally real field, the least number s(F) so that −1 is a sum of s(F) squares is always a power of 2 (cf. [109]). Interest in and problems about other arithmetic field invariants have also played a role in the development of the theory.

The nondegenerate even-dimensional symmetric bilinear forms determine an ideal I(F) in the Witt ring of F, called the fundamental ideal. Its powers I^{n}(F) := (I(F))^{n}, each generated by appropriate Pfister forms, give an important filtration of W (F). The problem then arises: What ring theoretic properties respect this filtration? From W (F) one also forms the graded ring GW (F) associated to I(F) and asks the same question.

Using Matsumoto’s presentation of K_{2}(F) of a field (cf. [98]), Milnor gave an ad hoc definition of a graded ring K_{∗}(F) := ⨁_{n≥0} K_{n}(F) of a field in [106]. From the viewpoint of Galois cohomology, this was of great interest as there is a natural map, called the norm residue map, from K_{n}(F) to the Galois cohomology group H^{n}(Γ_{F} , μ_{m}^{⊗n}) where Γ_{F} is the absolute Galois group of F and m is relatively prime to the characteristic of F. For the case m = 2, Milnor conjectured this map to be an epimorphism with kernel 2K_{n}(F) for all n. Voevodsky proved this conjecture in [136]. Milnor also related his algebraic K-ring of a field to quadratic form theory by asking if GW (F) and K_{∗}(F)/2K_{∗}(F) are isomorphic. This was solved in the affirmative by Orlov, Vishik, and Voevodsky in [107]. Assuming these results, one can answer some of the questions that have arisen about the filtration of W (F) induced by the fundamental ideal.

In this book, we do not restrict ourselves to fields of characteristic different from 2. Historically the cases of fields of characteristic different from 2 and 2 have been studied separately. Usually the case of characteristic different from 2 is investigated first. In this book, we shall give characteristic free proofs whenever possible. This means that the study of symmetric bilinear forms and the study of quadratic forms must be done separately, then interrelated. We not only present the classical theory characteristic free but we also include many results not proven in any text as well as some previously unpublished results to bring the classical theory up to date.

We shall also take a more algebraic geometric viewpoint than has historically been done. Indeed, the final two parts of the book will be based on such a viewpoint. In our characteristic free approach, this means a firmer focus on quadratic forms which have nice geometric objects attached to them rather than on bilinear forms. We do this for a variety of reasons.

First, one can associate to a quadratic form a number of algebraic varieties: the quadric of isotropic lines in a projective space and, more generally, for an integer i > 0, the variety of isotropic subspaces of dimension i. More importantly, basic properties of quadratic forms can be reformulated in terms of the associated varieties: a quadratic form is isotropic if and only if the corresponding quadric has a rational point. A nondegenerate quadratic form is hyperbolic if and only if the variety of maximal totally isotropic subspaces has a rational point.

Not only are the associated varieties important but also the morphisms between them. Indeed, if φ is a quadratic form over F and L/F a finitely generated field extension, then there is a variety Y over F with function field L, and the form φ is isotropic over L if and only if there is a rational morphism from Y to the quadric of φ.

Working with correspondences rather than just rational morphisms adds further depth to our study, where we identify morphisms with their graphs. Working with these leads to the category of Chow correspondences. This provides greater flexibility because we can view correspondences as elements of Chow groups and apply the rich machinery of that theory: pull-back and push-forward homomorphisms, Chern classes of vector bundles, and Steenrod operations. For example, suppose we wish to prove that a property A of quadratic forms implies a property B. We translate the properties A and B to “geometric” properties A′ and B′ for the existence of certain cycles on certain varieties. Starting with cycles satisfying A′ we can then attempt to apply the operations over the cycles as above to produce cycles satisfying B′.

All the varieties listed above are projective homogeneous varieties under the action of the orthogonal group or special orthogonal group of φ, i.e., the orthogonal group acts transitively on the varieties. It is not surprising that the properties of quadratic forms are reflected in the properties of the special orthogonal groups. For example, if φ is of dimension 2n (with n ≥ 2) or 2n + 1 (with n ≥ 1), then the special orthogonal group is a semisimple group of type D_{n} or B_{n}. The classification of semisimple groups is characteristic free. This explains why most important properties of quadratic forms hold in all characteristics.

Unfortunately, bilinear forms are not “geometric”. We can associate varieties to a bilinear form, but it would be a variety of the associated quadratic form. Moreover, in characteristic 2 the automorphism group of a bilinear form is not semisimple.

In this book we sometimes give several proofs of the same results — one is classical, another is geometric. (This can be the same proof, but written in geometric language.) For example, this is done for Springer’s theorem and the Separation Theorem.

The first part of the text will derive classical results under this new setting. It is self-contained, needing minimal prerequisites except for Chapter VII. In this chapter we shall assume the results of Voevodsky in [136] and Orlov-Vishik-Voevodsky in [107] for fields of characteristic not 2, and Kato in [78] for fields of characteristic 2 on the solution for the analog of the Milnor Conjecture in algebraic K-theory. We do give new proofs for the case n = 2.

Prerequisites for the second two parts of the text will be more formidable. A reasonable background in algebraic geometry will be assumed. For the convenience of the reader appendices have been included as an aid. Unfortunately, we cannot give details of [136] or [107] as it would lead us away from the methods at hand.

The first part of this book covers the “classical” theory of quadratic forms, i.e., without heavy use of algebraic geometry, bringing it up to date. As the characteristic of a field is not deemed to be different from 2, this necessitates a bifurcation of the theory into the theory of symmetric bilinear forms and the theory of quadratic forms. The introduction of these subjects is given in the first two chapters.

Chapter I investigates the foundations of the theory of symmetric bilinear forms over a field F. Two major consequences of dealing with arbitrary characteristic are that such forms may not be diagonalizable and that nondegenerate isotropic planes need not be hyperbolic. With this taken into account, standard Witt theory, to the extent possible, is developed. In particular, Witt decomposition still holds, so that the Witt ring can be constructed in the usual way as well as the classical group presentation of the Witt ring W (F). This presentation is generalized to the fundamental ideal I(F) of even-dimensional forms in W (F) and then to the the second power I^{2}(F) of I(F), a theme returned to in Chapter VII. The Stiefel-Whitney invariants of bilinear forms are introduced along with their relationship with the invariants ē_{n} : I^{n}(F)/I^{n+1}(F) → K_{n}(F)/2K_{n+1}(F) for n = 1, 2. The theory of bilinear Pfister forms is introduced and some basic properties developed. Following [32], we introduce chain p-equivalence and linkage of Pfister forms as well as introducing annihilators of Pfister forms in the Witt ring.

Chapter II investigates the foundations of the theory of quadratic forms over a field F. Because of the arbitrary characteristic assumption on the field F, the definition of nondegenerate must be made more carefully, and quadratic forms are far from having orthogonal bases in general. Much of Witt theory, however, goes through as the Witt Extension Theorem holds for quadratic forms under fairly weak assumptions, hence Witt Decomposition. The Witt group I_{q}(F) of nondegenerate even-dimensional quadratic forms is defined and shown to be a W (F)-module. The theory of quadratic Pfister forms is introduced and some results analogous to that of the bilinear case are introduced. Moreover, cohomological invariants of quadratic Pfister forms are introduced and some preliminary results about them and their extension to the appropriate filtrant of the Witt group of quadratic forms are discussed. In addition, the classical quadratic form invariants, discriminant and Clifford invariant, are defined.

Chapter III begins the utilization of function field techniques in the study of quadratic forms, all done without restriction of characteristic. The classical Cassels-Pfister theorem is established. Values of anisotropic quadratic and bilinear forms over a polynomial ring are investigated, special cases being the representation of one form as a subform of another and various norm principles due to Knebusch (cf. [82]). To investigate norm principles of similarity factors due to Scharlau (cf. [119]), quadratic forms over valuation rings and transfer maps are introduced.

Chapter IV introduces algebraic geometric methods, i.e., looking at the theory under the base extension of the function field of a fixed quadratic form. In particular, the notion of domination of one form by another is introduced where an anisotropic quadratic form φ is said to dominate an anisotropic quadratic form ψ (both of dimension of at least two) if φ_{F (ψ)} is isotropic. The geometric properties of Pfister forms are developed, leading to the Arason-Pfister Hauptsatz that nonzero anisotropic quadratic (respectively, symmetric bilinear) forms in I_{q}^{n}(F) (respectively, I^{n}(F)) are of dimension at least 2^{n} and its application to linkage of Pfister forms. Knebusch’s generic tower of an anisotropic quadratic form is introduced and the W (F)-submodules J_{n}(F) of I_{q}(F) are defined by the notion of degree. These submodules turn out to be precisely the corresponding I_{q}^{n}(F) (to be shown in Chapter VII). Hoffmann’s Separation Theorem that if φ and ψ are two anisotropic quadratic forms over F with dim φ ≤ 2^{n} < dim ψ for some n ≥ 0, then φ_{F (ψ)} is anisotropic is proven as well as Fitzgerald’s theorem characterizing quadratic Pfister forms. In addition, excellent forms and extensions are discussed. In particular, Arason’s result that the extension of a field by the function field of a nondegenerate 3-dimensional quadratic form is excellent is proven. The chapter ends with a discussion of central simple algebras over the function field of a quadric.

Chapter V studies symmetric bilinear and quadratic forms under field extensions. The chapter begins with the study of the structure of the Witt ring of a field F based on the work of Pfister. After dispensing with the nonformally real F, we turn to the study over a formally real field utilizing the theory of pythagorean fields and the pythagorean closure of a field, leading to the Local-Global Theorem of Pfister and its consequences for structure of the Witt ring over a formally real field. The total signature map from the Witt ring to the ring of continuous functions from the order space of a field to the integers is then carefully studied, in particular, the approximation of elements in this ring of functions by quadratic forms. The behavior of quadratic and bilinear forms under quadratic extensions (both separable and inseparable) is then investigated. The special case of the torsion of the Witt ring under such extensions is studied. A detailed investigation of torsion Pfister forms is begun, leading to the theorem of Krüskemper which implies that if K/F is a quadratic field extension with I^{n}(K) = 0, then I^{n}(F) is torsion-free.

Chapter VI studies u-invariants, their behavior under field extensions, and values that they can take. Special attention is given to the case of formally real fields.

Chapter VII establishes consequences of the result of Orlov-Vishik-Voevodsky in [107] which we assume in this chapter. In particular, answers and generalizations of results from the previous chapters are established. For fields of characteristic not 2, the ideals I^{n}(F) and J_{n}(F) are shown to be identical. The annihilators of Pfister forms in the Witt ring are shown to filter through the I^{n}(F), i.e., the intersection of such annihilators and I^{n}(F) are generated by Pfister forms in the intersection. A consequence is that torsion in I^{n}(F) is generated by torsion n-fold Pfister forms, solving a conjecture of Lam. A presentation for the group structure of the I^{n}(F)’s is determined, generalizing that given for I^{2}(F) in Chapter I. Finally, it is shown that if K/F is a finitely generated field extension of transcendence degree m, then I^{n}(K) torsion-free implies the same for I^{n−m}(F).

In Chapter VIII, we give a new elementary proof of the theorem in [100] that the second cohomological invariant is an isomorphism in the case that the characteristic of the field is different from 2 (the case of characteristic 2 having been done in Chapter II). This is equivalent to the degree two case of the Milnor Conjecture in [106] stating that the norm residue homomorphism

h_{F}^{n} : K_{n}(F)/2K_{n}(F) → H^{n}(F, ℤ/2ℤ)

is an isomorphism for every integer n. The Milnor Conjecture was proven in full by V. Voevodsky in [136]. Unfortunately, the scope of this book does not allow us to prove this beautiful result as the proof requires motivic cohomology and Steenrod operations developed by Voevodsky. In Chapter VIII, we give an “elementary” proof of the degree two case of the Milnor Conjecture that does not rely either on a specialization argument or on higher K-theory as did the original proof of this case in [100].

In the second part of the book, we develop the needed tools in algebraic geometry that will be applied in the third part. The main object studied in Part Two is the Chow group of algebraic cycles modulo rational equivalence on an algebraic scheme. Using algebraic cycles, we introduce the category of correspondences.

In Chapter IX (following the approach of [117] given by Rost), we develop the K-homology and K-cohomology theories of schemes over a field. This generalizes the Chow groups. We establish functorial properties of these theories (pull-back, push-forward, deformation and Gysin homomorphisms), introduce Euler and Chern classes of vector bundles, and prove basic results such as the Homotopy Invariance and Projective Bundle Theorems. We apply these results to Chow groups in the next chapter.

Chapter XI is devoted to the study of Steenrod operations on Chow groups modulo 2 over fields of characteristic not 2. Steenrod operations for motivic cohomology modulo a prime integer p of a scheme X were originally constructed by Voevodsky in [137]. The reduced power operations (but not the Bockstein operation) restrict to the Chow groups of X. An “elementary” construction of the reduced power operations modulo p on Chow groups (requiring equivariant Chow groups) was given by Brosnan in [20].

In Chapter XII, we introduce the notion of a Chow motive that is due to Grothendieck. Many (co)homology theories defined on the category of smooth complete varieties, such as Chow groups and more generally the K-(co)homology groups, take values in the category of abelian groups. But the category of smooth complete varieties itself does not have the structure of an additive category as we cannot add morphisms of varieties. The category of Chow motives, however, is an additive tensor category. This additional structure gives more flexibility when working with regular and rational morphisms.

In the third part of the book we apply algebraic geometric methods to the further study of quadratic forms. In Chapter XIII, we prove preliminary facts about algebraic cycles on quadrics and their powers. We also introduce shell triangles and diagrams of cycles, the basic combinatorial objects associated to a quadratic form. The corresponding pictures of these shell triangles simplify visualization of algebraic cycles and operations over the cycles.

In Chapter XIV, we study the Izhboldin dimension of smooth projective quadrics. It is defined as the integer

dim_{Izh}(X) := dim X − 𝔦_{1}(X) + 1,

where 𝔦_{1}(X) is the first Witt index of the quadric X. The Izhboldin dimension behaves better than the classical dimension with respect to splitting properties. For example, if X and Y are anisotropic smooth projective quadrics and Y is isotropic over the function field F (X), then dim_{Izh} X ≤ dim_{Izh} Y but dim X may be bigger than dim Y.

Chapter XV is devoted to applications of the Steenrod operations. The following problems are solved:

(1) All possible values of the first Witt index of quadratic forms are determined.

(2) All possible values of dimensions of anisotropic quadratic forms in I^{n}(F) are determined.

(3) It is shown that excellent forms have the smallest height among all quadratic forms of a given dimension.

In Chapter XVI, we study the variety of maximal isotropic subspaces of a quadratic forms. A discrete invariant J(φ) of a quadratic form φ is introduced. We also introduce the notion of canonical dimension and compute it for projective quadrics and varieties of totally isotropic subspaces.

In the last chapter we study motives of smooth projective quadrics in the category of correspondences and motives.

This book could not have been written without the help and encouragement of many of our friends, collaborators, and students nor the many researchers whose work was essential to producing this volume. These are too numerous to be listed here individually. We do wish to mention those who gave us valuable advice in the preparation and writing of this tome and those who helped proofread the manuscript: Jón Arason, Ricardo Baeza, Alex Boisvert, Detlev Hoffmann, Bryant Mathews, Claus Schubert, Jean-Pierre Tignol, Alexander Vishik, Maksim Zhykhovich. In addition, we would like to thank the referees who made valuable suggestions to improve the manuscript and Sergei Gelfand and the American Mathematical Society for its encouragement. It also gives us great pleasure to thank the National Science Foundation (grant DMS 0652316), as well as the Collaborative Research Centre 701 of the Bielefeld University, École Polytechnique Fédérale de Lausanne, Institute for Advanced Study in Princeton (The James D. Wolfensohn Fund and The Ellentuck Fund), Institut des Hautes Études Scientifiques, Institut Universitaire de France, and Max-Planck-Institut für Mathematik in Bonn, for their generous support. Finally, we wish to thank our families who put up with us through the long process of bringing this volume to fruition.

Part 1

Classical theory of symmetric bilinear forms and quadratic forms

CHAPTER I Bilinear Forms

1. Foundations

The study of (n × n)-matrices over a field F leads to various classification problems. Of special interest is to classify alternating and symmetric matrices. If A and B are two such matrices, we say that they are congruent if A = P^{t}BP for some invertible matrix P. For example, it is well-known that symmetric matrices are diagonalizable if the characteristic of F is different from 2. So the problem of classifying matrices up to congruence reduces to the study of a congruence class of a matrix in this case. The study of alternating and symmetric bilinear forms over an arbitrary field is the study of this problem in a coordinate-free approach. Moreover, we shall, whenever possible, give proofs independent of characteristic. In this section, we introduce the definitions and notation needed throughout the text and prove that we have a Witt Decomposition Theorem (cf. Theorem 1.27 below) for such forms. As we make no assumption on the characteristic of the underlying field, this makes the form of this theorem more delicate.

Definition 1.1. Let V be a finite dimensional vector space over a field F. A bilinear form on V is a map 𝔟 : V × V → F satisfying for all v, v′, w, w′ ∈ V and c ∈ F,

𝔟(v + v′, w) = 𝔟(v, w) + 𝔟(v′, w),
𝔟(v, w + w′) = 𝔟(v, w) + 𝔟(v, w′),
𝔟(cv, w) = c𝔟(v, w) = 𝔟(v, cw).

The bilinear form is called symmetric if 𝔟(v, w) = 𝔟(w, v) for all v, w ∈ V and is called alternating if 𝔟(v, v) = 0 for all v ∈ V. If 𝔟 is an alternating form, expanding 𝔟(v + w, v + w) shows that 𝔟 is skew symmetric, i.e., that 𝔟(v, w) = −𝔟(w, v) for all v, w ∈ V. In particular, every alternating form is symmetric if char F = 2. We call dim V the dimension of the bilinear form and also write it as dim 𝔟. We write 𝔟 is a bilinear form over F if 𝔟 is a bilinear form on a finite dimensional vector space over F and denote the underlying space by V_{𝔟}.

Let V^{∗} := Hom_{F} (V, F) denote the dual space of V. A bilinear form 𝔟 on V is called nondegenerate if l : V → V^{∗} defined by v ↦ l_{v} : w ↦ 𝔟(v, w) is an isomorphism. An isometry f : 𝔟_{1} → 𝔟_{2} between two bilinear forms 𝔟_{i}, i = 1, 2, is a linear isomorphism f : V_{𝔟_{1}} → V_{𝔟_{2}} such that 𝔟_{1}(v, w) = 𝔟_{2}(f(v), f(w)) for all v, w ∈ V_{𝔟_{1}}. If such an isometry exists, we write 𝔟_{1} ≃ 𝔟_{2} and say that 𝔟_{1} and 𝔟_{2} are isometric.

Let 𝔟 be a bilinear form on V. Let {v_{1}, …, v_{n}} be a basis for V. Then 𝔟 is determined by the matrix (𝔟(v_{i}, v_{j})) and the form is nondegenerate if and only if (𝔟(v_{i}, v_{j})) is invertible. Conversely, any matrix B in the n × n matrix ring 𝕄_{n}(F) determines a bilinear form based on V. If 𝔟 is symmetric (respectively, alternating), then the associated matrix is symmetric (respectively, alternating where a square matrix (a_{ij}) is called alternating if a_{ij} = −a_{ji} and a_{ii} = 0 for all i, j). Let 𝔟 and 𝔟′ be two bilinear forms with matrices B and B′ relative to some bases. Then 𝔟 ≃ 𝔟′ if and only if B′ = A^{t}BA for some invertible matrix A, i.e., the matrices B′ and B are congruent. As det B′ = det B · (det A)^{2} and det A ≠ 0, the determinant of B′ coincides with the determinant of B up to squares. We define the determinant of a nondegenerate bilinear form 𝔟 by det 𝔟 := det B · F^{×2} in F^{×}/F^{×2}, where B is a matrix representation of 𝔟 and F^{×} is the multiplicative group of F (and more generally, R^{×} denotes the unit group of a ring R). So the det is an invariant of the isometry class of a nondegenerate bilinear form.

The set Bil(V) of bilinear forms on V is a vector space over F. The space Bil(V) contains the subspaces Alt(V) of alternating forms on V and Sym(V) of symmetric bilinear forms on V. The correspondence of bilinear forms and matrices given above defines a linear isomorphism Bil(V) → 𝕄_{dim V} (F). If 𝔟 ∈ Bil(V), then 𝔟 − 𝔟^{t} is alternating where the bilinear form 𝔟^{t} is defined by 𝔟^{t}(v, w) = 𝔟(w, v) for all v, w ∈ V. Since every alternating n × n matrix is of the form B − B^{t} for some B, the linear map Bil(V) → Alt(V) given by 𝔟 ↦ 𝔟 − 𝔟^{t} is surjective. Therefore, we have an exact sequence of vector spaces

(1.2)  0 → Sym(V) → Bil(V) → Alt(V) → 0.

Exercise 1.3. Construct natural isomorphisms

Bil(V) ≃ (V ⊗_{F} V)^{∗} ≃ V^{∗} ⊗_{F} V^{∗}, Sym(V) ≃ S^{2}(V)^{∗},
Alt(V) ≃ Λ^{2}(V)^{∗} ≃ Λ^{2}(V^{∗})

and show that the exact sequence (1.2) is dual to the standard exact sequence

0 → Λ^{2}(V) → V ⊗_{F} V → S^{2}(V) → 0

where Λ^{2}(V) is the exterior square of V and S^{2}(V) is the symmetric square of V.

If 𝔟, 𝔠 ∈ Bil(V), we say the two bilinear forms 𝔟 and 𝔠 are similar if 𝔟 ≃ a𝔠 for some a ∈ F^{×}.

Let V be a finite dimensional vector space over F and let λ = ±1. Define the hyperbolic λ-bilinear form of V to be the form ℍ_{λ}(V) = 𝔟_{ℍ_{λ}} on V ⊕ V^{∗} defined by

𝔟_{ℍ_{λ}}(v_{1} + f_{1}, v_{2} + f_{2}) := f_{1}(v_{2}) + λf_{2}(v_{1})

for all v_{1}, v_{2} ∈ V and f_{1}, f_{2} ∈ V^{∗}. If λ = 1, the form ℍ_{λ}(V) is a symmetric bilinear form and if λ = −1, it is an alternating bilinear form. A bilinear form 𝔟 is called a hyperbolic bilinear form if 𝔟 ≃ ℍ_{λ}(W) for some finite dimensional F -vector space W and some λ = ±1. The hyperbolic form ℍ_{λ}(F) is called the hyperbolic plane and denoted ℍ_{λ}. It has the matrix representation

(\begin{matrix}0 & 1 \\ λ & 0 \end{matrix})

in the appropriate basis. If 𝔟 ≃ ℍ_{λ}, then 𝔟 has the above matrix representation in some basis {e, f} of V_{𝔟}. We call e, f a hyperbolic pair. Hyperbolic forms are nondegenerate.

Let 𝔟 be a bilinear form on V and W ⊂ V a subspace. The restriction of 𝔟 to W is a bilinear form on W and is called a subform of 𝔟. We denote this form by 𝔟|_{W}.

1.A. Structure theorems for bilinear forms. Let 𝔟 be a symmetric or alternating bilinear form on V. We say v, w ∈ V are orthogonal if 𝔟(v, w) = 0. Let W, U ⊂ V be subspaces. Define the orthogonal complement of W by

W^{⊥} := {v ∈ V | 𝔟(v, w) = 0 for all w ∈ W }.

This is a subspace of V. We say W is orthogonal to U if W ⊂ U^{⊥}, equivalently U ⊂ W^{⊥}. If V = W ⊕ U is a direct sum of subspaces with W ⊂ U^{⊥}, we write 𝔟 = 𝔟|_{W} ⊥ 𝔟|_{U} and say 𝔟 is the (internal) orthogonal sum of 𝔟|_{W} and 𝔟|_{U}. The subspace V^{⊥} is called the radical of 𝔟 and denoted by rad 𝔟. The form 𝔟 is nondegenerate if and only if rad 𝔟 = 0.

If K/F is a field extension, let V_{K} := K ⊗_{F} V, a vector space over K. We have the standard embedding V → V_{K} by v ↦ 1 ⊗ v. Let 𝔟_{K} denote the extension of 𝔟 to V_{K}, so 𝔟_{K}(a ⊗ v, c ⊗ w) = ac𝔟(v, w) for all a, c ∈ K and v, w ∈ V. The form 𝔟_{K} is of the same type as 𝔟. Moreover, rad 𝔟_{K} = (rad 𝔟)_{K}, hence 𝔟 is nondegenerate if and only if 𝔟_{K} is nondegenerate.

Let : V → \overline{V} := V/ rad 𝔟 be the canonical epimorphism. Define \overline{𝔟} to be the bilinear form on \overline{V} determined by \overline{𝔟}(\overline{v_{1}}, \overline{v_{2}}) := 𝔟(v_{1}, v_{2}) for all v_{1}, v_{2} ∈ V. Then \overline{𝔟} is a nondegenerate bilinear form of the same type as 𝔟. Note also that if f : 𝔟_{1} → 𝔟_{2} is an isometry of symmetric or alternative bilinear forms, then f(rad 𝔟_{1}) = rad 𝔟_{2}.

We have:

Lemma 1.4. Let 𝔟 be a symmetric or alternating bilinear form on V. Let W be any subspace of V such that V = rad 𝔟 ⊕ W. Then 𝔟|_{W} is nondegenerate and

𝔟 = 𝔟|_{rad 𝔟} ⊥ 𝔟|_{W} = 0|_{rad 𝔟} ⊥ 𝔟|_{W}

with 𝔟|_{W} ≃ \overline{𝔟}, the form induced on V/ rad 𝔟. In particular, 𝔟|_{W} is unique up to isometry.

The lemma above shows that it is sufficient to classify nondegenerate bilinear forms. In general, if 𝔟 is a symmetric or alternating bilinear form on V and W ⊂ V is a subspace, then we have an exact sequence of vector spaces

0 → W^{⊥} → V ⟶^{l_{W}} W^{∗},

where l_{W} is defined by v ↦ l_{v}|_{W} : x ↦ 𝔟(v, x). Hence dim W^{⊥} ≥ dim V − dim W. It is easy to determine when this is an equality.

Proposition 1.5. Let 𝔟 be a symmetric or alternating bilinear form on V. Let W be any subspace of V. Then the following are equivalent:

(1) W ∩ rad 𝔟 = 0.

(2) l_{W} : V → W^{∗} is surjective.

(3) dim W^{⊥} = dim V − dim W.

Proof. (1) holds if and only if the map l_{W}^{∗} : W → V^{∗} is injective, if and only if the map l_{W} : V → W^{∗} is surjective, and if and only if (3) holds. □

Note that the conditions (1)-(3) hold if either 𝔟 or 𝔟|_{W} is nondegenerate.

A key observation is:

Proposition 1.6. Let 𝔟 be a symmetric or alternating bilinear form on V. Let W be a subspace such that 𝔟|_{W} is nondegenerate. Then 𝔟 = 𝔟|_{W} ⊥ 𝔟|_{W^{⊥}}. In particular, if 𝔟 is also nondegenerate, then so is 𝔟|_{W^{⊥}}.

Proof. By Proposition 1.5, dim W^{⊥} = dim V − dim W, hence V = W ⊕ W^{⊥}. The result follows. □

Corollary 1.7. Let 𝔟 be a symmetric bilinear form on V. Let v ∈ V satisfy 𝔟(v, v) ≠ 0. Then 𝔟 = 𝔟|_{F v} ⊥ 𝔟_{(F v)^{⊥}}.

Let 𝔟_{1} and 𝔟_{2} be two symmetric or alternating bilinear forms on V_{1} and V_{2} respectively. Then their external orthogonal sum, denoted by 𝔟_{1} ⊥ 𝔟_{2}, is the form on V_{1} ⊕ V_{2} given by

(𝔟_{1} ⊥ 𝔟_{2})((v_{1}, v_{2}), (w_{1}, w_{2})) := 𝔟_{1}(v_{1}, w_{1}) + 𝔟_{2}(v_{2}, w_{2})

for all v_{i}, w_{i} ∈ V_{i}, i = 1, 2.

If n is a nonnegative integer and 𝔟 is a symmetric or alternating bilinear form over F, abusing notation, we let

n𝔟 := \underbrace{𝔟 ⊥ ⋯ ⊥ 𝔟}_{n} .

In particular, if n is a nonnegative integer, we do not interpret n𝔟 with n viewed in the field.

For example, ℍ_{λ}(V) ≃ nℍ_{λ} for any n-dimensional vector space V over F. It is now easy to complete the classification of alternating forms.

Proposition 1.8. Let 𝔟 be a nondegenerate alternating form on V. Then dim V = 2n for some n and 𝔟 ≃ nℍ_{−1}, i.e., 𝔟 is hyperbolic.

Proof. Let 0 ≠ v ∈ V. Then there exists w ∈ V such that 𝔟(v, w) = a ≠ 0. Replacing w by a^{−1}w, we see that v, w is a hyperbolic pair in the space W = F v ⊕ F w, so 𝔟|_{W} is a hyperbolic subform of 𝔟, in particular, nondegenerate. Therefore, 𝔟 = 𝔟|_{W} ⊥ 𝔟|_{W^{⊥}} by Proposition 1.6. The result follows by induction on dim 𝔟. □

The proof shows that every nondegenerate alternating form 𝔟 on V has a symplectic basis, i.e., a basis {v_{1}, …, v_{2n}} for V satisfying 𝔟(v_{i}, v_{n+i}) = 1 for all i ∈ [1, n] := {i ∈ ℤ | 1 ≤ i ≤ n} and 𝔟(v_{i}, v_{j}) = 0 if i ≤ j and j ≠ n + i.

We turn to the classification of the isometry type of symmetric bilinear forms. By Lemma 1.4, Corollary 1.7, and induction, we therefore have the following:

Corollary 1.9. Let 𝔟 be a symmetric bilinear form on V. Then

𝔟 = 𝔟|_{rad 𝔟} ⊥ 𝔟|_{V_{1}} ⊥ ⋯ ⊥ 𝔟|_{V_{n}} ⊥ 𝔟|_{W}

with V_{i} a 1-dimensional subspace of V and 𝔟|_{V_{i}} nondegenerate for all i ∈ [1, n] and 𝔟|_{W} a nondegenerate alternating subform on a subspace W of V.

If char F ≠ 2, then, in the corollary, 𝔟|_{W} is symmetric and alternating hence W = {0}. In particular, every symmetric bilinear form 𝔟 has an orthogonal basis, i.e., a basis {v_{1}, …, v_{n}} for V_{𝔟} satisfying 𝔟(v_{i}, v_{j}) = 0 if i ≠ j. The form is nondegenerate if and only if 𝔟(v_{i}, v_{i}) ≠ 0 for all i.

If char F = 2, by Proposition 1.8, the alternating form 𝔟|_{W} in the corollary above has a symplectic basis and satisfies 𝔟|_{W} ≃ nℍ_{1} for some n.

Let a ∈ F. Denote the bilinear form on F given by 𝔟(v, w) = avw for all v, w ∈ F by ⟨a⟩_{b} or simply ⟨a⟩. In particular, ⟨a⟩ ≃ ⟨b⟩ if and only if a = b = 0 or aF^{×2} = bF^{×2} in F^{×}/F^{×2}. Denote

⟨a_{1}⟩ ⊥ ⋯ ⊥ ⟨a_{n}⟩ by ⟨a_{1}, …, a_{n}⟩_{b} or simply by ⟨a_{1}, …, a_{n}⟩.

We call such a form a diagonal form. A symmetric bilinear form 𝔟 isometric to a diagonal form is called diagonalizable. Consequently, 𝔟 is diagonalizable if and only if 𝔟 has an orthogonal basis. Note that det⟨a_{1}, …, a_{n}⟩ = a_{1} ⋯ a_{n}F^{×2} if a_{i} ∈ F^{×} for all i. Corollary 1.9 says that every bilinear form 𝔟 on V satisfies

𝔟 ≃ r⟨0⟩ ⊥ ⟨a_{1}, …, a_{n}⟩ ⊥ 𝔟′

with r = dim(rad 𝔟) and 𝔟′ an alternating form and a_{i} ∈ F^{×} for all i. In particular, if char F ≠ 2, then every symmetric bilinear form is diagonalizable.

Example 1.10. Let a, b ∈ F^{×}. Then ⟨1, a⟩ ≃ ⟨1, b⟩ if and only if aF^{×2} = det⟨1, a⟩ = det⟨1, b⟩ = bF^{×2}.

1.B. Values and similarities of bilinear forms. We study the values that a bilinear form can take as well as the similarity factors. We begin with some notation.

Definition 1.11. Let 𝔟 be a bilinear form on V over F. Let

D(𝔟) := {𝔟(v, v) | v ∈ V with 𝔟(v, v) ≠ 0},

the set on nonzero values of 𝔟 and

G(𝔟) := {a ∈ F^{×} | a𝔟 ≃ 𝔟},

a group called the group of similarity factors of 𝔟. Also set

D̃(𝔟) := D(𝔟) ∪ {0}.

We say that elements in D̃(𝔟) are represented by 𝔟.

For example, G(ℍ_{1}) = F^{×}. A symmetric bilinear form is called round if G(𝔟) = D(𝔟). In particular, if 𝔟 is round, then D(𝔟) is a group.

Remark 1.12. If 𝔟 is a symmetric bilinear form and a ∈ D(𝔟), then 𝔟 ≃ ⟨a⟩ ⊥ 𝔠 for some symmetric bilinear form 𝔠 by Corollary 1.7.

Lemma 1.13. Let 𝔟 be a bilinear form. Then

D(𝔟) · G(𝔟) ⊂ D(𝔟).

In particular, if 1 ∈ D(𝔟), then G(𝔟) ⊂ D(𝔟).

Proof. Let a ∈ G(𝔟) and b ∈ D(𝔟). Let λ : 𝔟 → a𝔟 be an isometry and v ∈ V_{𝔟} satisfy b = 𝔟(v, v). Then 𝔟(λ(v), λ(v)) = a𝔟(v, v) = ab. □

Example 1.14. Let K = F [t]/(t^{2} − a) with a ∈ F, where F [t] is the polynomial ring over F. So K = F ⊕ F θ as a vector space over F where θ denotes the class of t in K. If z = x + yθ with x, y ∈ F, write \overline{z} = x − yθ. Let s : K → F be the F -linear functional defined by s(x + yθ) = x. Then 𝔟 defined by 𝔟(z_{1}, z_{2}) = s(z_{1}\overline{z}_{2}) is a binary symmetric bilinear form on K. Let N(z) = z\overline{z} for z ∈ K. Then D(𝔟) = {N(z) ≠ 0 | z ∈ K} = {N(z) | z ∈ K^{×}}. If z ∈ K, then λ_{z} : K → K given by w → zw is an F -linear isomorphism if and only if N(z) ≠ 0. Suppose that λ_{z} is an F -isomorphism. As

𝔟(λ_{z}z_{1}, λ_{z}z_{2}) = 𝔟(zz_{1}, zz_{2}) = N(z)s(z_{1}\overline{z}_{2}) = N(z)𝔟(z_{1}, z_{2}),

we have an isometry N(z)𝔟 ≃ 𝔟 for all z ∈ K^{×}. In particular, 𝔟 is round. Computing 𝔟 on the orthogonal basis {1, θ} for K shows that 𝔟 is isometric to the bilinear form ⟨1, −a⟩. If a ∈ F^{×}, then 𝔟 ≃ ⟨1, −a⟩ is nondegenerate.

Remark 1.15. (1) Let 𝔟 be a binary symmetric bilinear form on V. Suppose there exists a basis {v, w} for V satisfying 𝔟(v, v) = 0, 𝔟(v, w) = 1, and 𝔟(w, w) = a ≠ 0. Then 𝔟 is nondegenerate as the matrix corresponding to 𝔟 in this basis, is invertible. Moreover, {w, −av + w} is an orthogonal basis for V and, using this basis, we see that 𝔟 ≃ ⟨a, −a⟩.

(2) Suppose that char F ≠ 2. Let 𝔟 = ⟨a, −a⟩ with a ∈ F^{×} and {e, g} an orthogonal basis for V_{𝔟} satisfying a = 𝔟(e, e) = −𝔟(f, f). Evaluating on the basis \{e + f, \frac{1}{2a}(e − f)\} shows that 𝔟 ≃ ℍ_{1}. In particular, ⟨a, −a⟩ ≃ ℍ_{1} for all a ∈ F^{×}. Moreover, ⟨a, −a⟩ ≃ ℍ_{1} is round and universal, where a nondegenerate symmetric bilinear form 𝔟 is called universal if D(𝔟) = F^{×}.

(3) Suppose that char F = 2. As ℍ_{1} = ℍ_{−1} is alternating while ⟨a, a⟩ is not, ⟨a, a⟩ ≄ ℍ_{1} for any a ∈ F^{×}. Moreover, ℍ_{1} is not round since D(ℍ_{1}) = ∅. As D(⟨a, a⟩) = D(⟨a⟩) = aF^{×2}, we have G(⟨a, a⟩) = F^{×2} by Lemma 1.13. In particular, ⟨a, a⟩ is round if and only if a ∈ F^{×2}, and ⟨a, a⟩ ≃ ⟨b, b⟩ if and only if aF^{×2} ≃ bF^{×2}.

(4) Witt Cancellation holds if char F ≠ 2, i.e., if there exists an isometry of symmetric bilinear forms 𝔟 ⊥ 𝔟′ ≃ 𝔟 ⊥ 𝔟′′ over F with 𝔟 nondegenerate, then 𝔟′ ≃ 𝔟′′. (Cf. Theorem 8.4 below.) If char F = 2, this is false in general. For example,

⟨1, 1, −1⟩ ≃ ⟨1⟩ ⊥ ℍ_{1}

over any field. Indeed if 𝔟 is 3-dimensional on V and V has an orthogonal basis {e, f, g} with 𝔟(e, e) = 1 = 𝔟(f, f) and 𝔟(g, g) = −1, then the right hand side arises from the basis {e + f + g, e + g, −f − g}. But by (3), ⟨1, −1⟩ ≄ ℍ_{1} if char F = 2. Multiplying the equation above by any a ∈ F^{×}, we also have

(1.16)  ⟨a, a, −a⟩ ≃ ⟨a⟩ ⊥ ℍ_{1}.

Proposition 1.17. Let 𝔟 be a symmetric bilinear form. If D(𝔟) ≠ ∅, then 𝔟 is diagonalizable. In particular, a nonzero symmetric bilinear form is diagonalizable if and only if it is not alternating.

Proof. If a ∈ D(𝔟), then

𝔟 ≃ ⟨a⟩ ⊥ 𝔟_{1} ≃ ⟨a⟩ ⊥ rad 𝔟_{1} ⊥ 𝔠_{1} ⊥ 𝔠_{2}

with 𝔟_{1} a symmetric bilinear form by Corollary 1.7, and 𝔠_{1} a nondegenerate diagonal form, and 𝔠_{2} a nondegenerate alternating form by Corollary 1.9. By the remarks following Corollary 1.9, we have 𝔠_{2} = 0 if char F ≠ 2 and 𝔠_{2} = mℍ_{1} for some integer m if char F = 2. By (1.16), we conclude that 𝔟 is diagonalizable in either case.

If 𝔟 is not alternating, then D(𝔟) ≠ ∅, hence 𝔟 is diagonalizable. Conversely, if 𝔟 is diagonalizable, it cannot be alternating as it is not the zero form. □

Corollary 1.18. Let 𝔟 be a symmetric bilinear form over F. Then 𝔟 ⊥ ⟨1⟩ is diagonalizable.

Let 𝔟 be a symmetric bilinear form on V. A vector v ∈ V is called anisotropic if 𝔟(v, v) ≠ 0 and isotropic if v ≠ 0 and 𝔟(v, v) = 0. We call 𝔟 anisotropic if there are no isotropic vectors in V and isotropic otherwise.

Corollary 1.19. Every anisotropic bilinear form is diagonalizable.

Note that an anisotropic symmetric bilinear form is nondegenerate as its radical is trivial.

Example 1.20. Let F be a quadratically closed field, i.e., every element in F is a square. Then, up to isometry, 0 and ⟨1⟩ are the only anisotropic forms over F. In particular, this applies if F is algebraically closed.

An anisotropic form may not be anisotropic under base extension. However, we do have:

Lemma 1.21. Let 𝔟 be an anisotropic bilinear form over F. If K/F is purely transcendental, then 𝔟_{K} is anisotropic.

Proof. First suppose that K = F (t), the field of rational functions over F in the variable t. Suppose that 𝔟_{F (t)} is isotropic. Then there exist a vector 0 ≠ v ∈ V_{𝔟_{F (t)}} such that 𝔟_{F (t)}(v, v) = 0. Multiplying by an appropriate nonzero polynomial, we may assume that v ∈ F [t] ⊗_{F} V. Write v = v_{0} + t ⊗ v_{1} + ⋯ + t^{n} ⊗ v_{n} with v_{1}, …, v_{n} ∈ V and v_{n} ≠ 0. As the t^{2n} coefficient 𝔟(v_{n}, v_{n}) of 𝔟(v, v) must vanish, v_{n} is an isotropic vector of 𝔟, a contradiction.

If K/F is finitely generated, then the result follows by induction on the transcendence degree of K over F. In the general case, if 𝔟_{K} is isotropic there exists a finitely generated purely transcendental extension K_{0} of F in K with 𝔟_{K_{0}} isotropic, a contradiction. □

1.C. Metabolic bilinear forms. Let 𝔟 be a symmetric bilinear form on V. A subspace W ⊂ V is called a totally isotropic subspace of 𝔟 if 𝔟|_{W} = 0, i.e., if W ⊂ W^{⊥}. If 𝔟 is isotropic, then it has a nonzero totally isotropic subspace. Suppose that 𝔟 is nondegenerate and W is a totally isotropic subspace. Then dim W + dim W^{⊥} = dim V by Proposition 1.5, hence dim W ≤ \frac{1}{2} dim V. We say that W is a lagrangian for b if we have an equality dim W = \frac{1}{2} dim V, equivalently W^{⊥} = W. A nondegenerate symmetric bilinear form is called metabolic if it has a lagrangian. In particular, every metabolic form is even-dimensional. Clearly, an orthogonal sum of metabolic forms is metabolic.

Example 1.22. (1) Symmetric hyperbolic forms are metabolic.

(2) The form 𝔟 ⊥ (−𝔟) is metabolic if 𝔟 is any nondegenerate symmetric bilinear form.

(3) A 2-dimensional metabolic space is nothing but a nondegenerate isotropic plane. A metabolic plane is therefore either isomorphic to ⟨a, −a⟩ for some a ∈ F^{×} or to the hyperbolic plane ℍ_{1} by Remark 1.15. In particular, the determinant of a metabolic plane is −F^{×2}. If char F ≠ 2, then ⟨a, −a⟩ ≃ ℍ_{1} by Remark 1.15, so in this case, every metabolic plane is hyperbolic.

Lemma 1.23. Let 𝔟 be an isotropic nondegenerate symmetric bilinear form over V. Then every isotropic vector belongs to a 2-dimensional metabolic subform.

Proof. Suppose that 𝔟(v, v) = 0 with v ≠ 0. As 𝔟 is nondegenerate, there exists a u ∈ V such that 𝔟(u, v) ≠ 0. Then 𝔟|_{F v⊕F u} is metabolic. □

Corollary 1.24. Every metabolic form is an orthogonal sum of metabolic planes. In particular, if 𝔟 is a metabolic form over F, then det 𝔟 = (−1)^{\frac{dim 𝔟}{2}} F^{×2}.

Proof. We induct on the dimension of a metabolic form 𝔟. Let W ⊂ V = V_{𝔟} be a lagrangian. By Lemma 1.23, a nonzero vector v ∈ W belongs to a metabolic plane P ⊂ V. It follows from Proposition 1.6 that 𝔟 = 𝔟|_{P} ⊥ 𝔟|_{P^{⊥}} and by dimension count that W ∩P^{⊥} is a lagrangian of 𝔟|_{P^{⊥}} as W cannot lie in P^{⊥}. By the induction hypothesis, 𝔟|_{P^{⊥}} is an orthogonal sum of metabolic planes. The second statement follows from Example 1.22(3). □

Corollary 1.25. If char F ≠ 2, the classes of metabolic and hyperbolic forms coincide. In particular, every isotropic nondegenerate symmetric bilinear form is universal.

Proof. This follows from Remark 1.15(2) and Lemma 1.23. □

Lemma 1.26. Let 𝔟 and 𝔟′ be two symmetric bilinear forms. If 𝔟 ⊥ 𝔟′ and 𝔟′ are both metabolic, then so is 𝔟.

Proof. By Corollary 1.24, we may assume that 𝔟′ is 2-dimensional. Let W be a lagrangian for 𝔟 ⊥ 𝔟′. Let p : W → V_{𝔟′} be the projection and W_{0} = Ker(p) = W ∩ V_{𝔟}. Suppose that p is not surjective. Then dim W_{0} ≥ dim W − 1, hence W_{0} is a lagrangian of 𝔟 and 𝔟 is metabolic.

So we may assume that p is surjective. Then dim W_{0} = dim W − 2. As 𝔟′ is metabolic, it is isotropic. Choose an isotropic vector v′ ∈ V_{𝔟′} and a vector w ∈ W such that p(w) = v′, i.e., w = v + v′ for some v ∈ V_{𝔟}. In particular, 𝔟(v, v) = (𝔟 ⊥ 𝔟′)(w, w) − 𝔟′(v′, v′) = 0. Since W_{0} ⊂ V_{𝔟}, we have v′ is orthogonal to W_{0}, hence v is also orthogonal to W_{0}. If we show that v′ ∉ W, then v ∉ W_{0} and W_{0} ⊕ F v is a lagrangian of 𝔟 and 𝔟 is metabolic.

So suppose v′ ∈ W. There exists v′′ ∈ V_{𝔟′} such that 𝔟′(v′, v′′) ≠ 0 as 𝔟′ is nondegenerate. Since p is surjective, there exists w′′ ∈ W with w′′ = u′′ + v′′ for some u′′ ∈ V_{𝔟}. As W is totally isotropic,

0 = (𝔟 ⊥ 𝔟′)(v′, w′′) = (𝔟 ⊥ 𝔟′)(v′, u′′ + v′′) = 𝔟′(v′, v′′),

a contradiction. □

1.D. Witt Theory. We have the following form of the classical Witt Decomposition Theorem (cf. [139]) for symmetric bilinear forms over a field of arbitrary characteristic.

Theorem 1.27 (Bilinear Witt Decomposition Theorem). Let 𝔟 be a nondegenerate symmetric bilinear form on V. Then there exist subspaces V_{1} and V_{2} of V such that 𝔟 = 𝔟|_{V_{1}} ⊥ 𝔟|_{V_{2}} with 𝔟|_{V_{1}} anisotropic and 𝔟|_{V_{2}} metabolic. Moreover, 𝔟|_{V_{1}} is unique up to isometry.

Proof. We prove existence of the decomposition by induction on dim 𝔟. If 𝔟 is isotropic, there is a metabolic plane P ⊂ V by Lemma 1.23. As 𝔟 = 𝔟|_{P} ⊥ 𝔟|_{P^{⊥}}, the proof of existence follows by applying the induction hypothesis to 𝔟|_{P^{⊥}}.

To prove uniqueness, assume that 𝔟_{1} ⊥ 𝔟_{2} ≃ 𝔟′_{1} ⊥ 𝔟′_{2} with 𝔟_{1} and 𝔟′_{1} both anisotropic and 𝔟_{2} and 𝔟′_{2} both metabolic. We show that 𝔟_{1} ≃ 𝔟′_{1}. The form

𝔟_{1} ⊥ (−𝔟′_{1}) ⊥ 𝔟_{2} ≃ 𝔟′_{1} ⊥ (−𝔟′_{1}) ⊥ 𝔟′_{2}

is metabolic, hence 𝔟_{1} ⊥ (−𝔟′_{1}) is metabolic by Lemma 1.26. Let W be a lagrangian of 𝔟_{1} ⊥ (−𝔟′_{1}). Since 𝔟_{1} is anisotropic, the intersection W ∩V_{𝔟_{1}} is trivial. Therefore, the projection W → V_{𝔟′_{1}} is injective and dim W ≤ dim 𝔟′_{1}. Similarly, dim W ≤ dim 𝔟_{1}. Consequently, dim 𝔟_{1} = dim W = dim 𝔟′_{1} and the projections p : W → V_{𝔟_{1}} and p′ : W → V_{𝔟′_{1}} are isomorphisms. Let w = v + v′ ∈ W, where v ∈ V_{𝔟_{1}} and v′ ∈ V_{𝔟′_{1}}. As

0 = (𝔟_{1} ⊥ (−𝔟′_{1}))(w, w) = 𝔟_{1}(v, v) − 𝔟′_{1}(v′, v′),

the isomorphism p′ ∘ p^{−1} : V_{𝔟_{1}} → V_{𝔟′_{1}} is an isometry between 𝔟_{1} and 𝔟′_{1}. □

Let 𝔟 = 𝔟|_{V_{1}} ⊥ 𝔟|_{V_{2}} be the decomposition of the nondegenerate symmetric bilinear form 𝔟 on V in the theorem. The anisotropic form 𝔟|_{V_{1}}, unique up to isometry, will be denoted by 𝔟_{an} and called the anisotropic part of 𝔟. Note that the metabolic form 𝔟|_{V_{2}} in Theorem 1.27 is not unique in general by Remark 1.15(4). However, its dimension is unique and even. Define the Witt index of 𝔟 to be 𝔦(𝔟) := (dim V_{2})/2 .

Remark 1.15(4) also showed that the Witt Cancellation Theorem does not hold for nondegenerate symmetric bilinear forms in characteristic 2. The obstruction is the metabolic forms. We have, however, the following:

Corollary 1.28 (Witt Cancellation). Let 𝔟, 𝔟_{1}, 𝔟_{2} be nondegenerate symmetric bilinear forms satisfying 𝔟_{1} ⊥ 𝔟 ≃ 𝔟_{2} ⊥ 𝔟. If 𝔟_{1} and 𝔟_{2} are anisotropic, then 𝔟_{1} ≃ 𝔟_{2}.

Proof. We have 𝔟_{1} ⊥ 𝔟 ⊥ (−𝔟) ≃ 𝔟_{2} ⊥ 𝔟 ⊥ (−𝔟) with 𝔟 ⊥ (−𝔟) metabolic. By Theorem 1.27, 𝔟_{1} ≃ 𝔟_{2}. □

2. The Witt and Witt-Grothendieck rings of symmetric bilinear forms

In this section, we construct the Witt ring. The orthogonal sum induces an additive structure on the isometry classes of symmetric bilinear forms. Defining the tensor product of symmetric bilinear forms (corresponding to the classical Kronecker product of matrices) turns this set of isometry classes into a semi-ring. The Witt Decomposition Theorem leads to a nice description of the Grothendieck ring in terms of isometry classes of anisotropic symmetric bilinear forms. The Witt ring W (F) is the quotient of this ring by the ideal generated by the hyperbolic plane.

Let 𝔟_{1} and 𝔟_{2} be symmetric bilinear forms over F. The tensor product of 𝔟_{1} and 𝔟_{2} is defined to be the symmetric bilinear form 𝔟 := 𝔟_{1} ⊗ 𝔟_{2} with underlying space V_{𝔟_{1}} ⊗_{F} V_{𝔟_{2}} with the form 𝔟 defined by

𝔟(v_{1} ⊗ v_{2}, w_{1} ⊗ w_{2}) = 𝔟_{1}(v_{1}, w_{1}) · 𝔟_{2}(v_{2}, w_{2})

for all v_{1}, w_{1} ∈ V_{𝔟_{1}} and v_{2}, w_{2} ∈ V_{𝔟_{2}}. For example, if a ∈ F, then ⟨a⟩ ⊗ 𝔟_{1} ≃ a𝔟_{1}.

Lemma 2.1. Let 𝔟_{1} and 𝔟_{2} be two nondegenerate bilinear forms over F. Then (1) 𝔟_{1} ⊥ 𝔟_{2} is nondegenerate.

(2) 𝔟_{1} ⊗ 𝔟_{2} is nondegenerate.

(3) ℍ_{1}(V) ⊗ 𝔟_{1} is hyperbolic for all finite dimensional vector spaces V.

Proof. (1), (2): Let V_{i} = V_{𝔟_{i}} for i = 1, 2. The 𝔟_{i} induce isomorphisms l_{i} : V_{i} → V_{i}^{∗} for i = 1, 2, hence 𝔟_{1} ⊥ 𝔟_{2} and 𝔟_{1} ⊗ 𝔟_{2} induce isomorphisms l_{1} ⊕ l_{2} : V_{1} ⊕ V_{2} → (V_{1} ⊕ V_{2})^{∗} and l_{1} ⊗ l_{2} : V_{1} ⊗_{F} V_{2} → (V_{1} ⊗_{F} V_{2})^{∗} respectively.

(3): Let {e, f} be a hyperbolic pair for ℍ_{1}. Then the linear map (F ⊕ F^{∗}) ⊗_{F} V_{1} → V_{1} ⊕V_{1}^{∗} induced by e⊗v ↦ v and f ⊗v ↦ l_{v} : w ↦ 𝔟(w, v) is an isomorphism and induces the isometry ℍ_{1} ⊗ 𝔟 → ℍ_{1}(V). □

It follows that the isometry classes of nondegenerate symmetric bilinear forms over F is a semi-ring under orthogonal sum and tensor product. The Grothendieck ring of this semi-ring is called the Witt-Grothendieck ring of F and denoted by Ŵ(F). (Cf. Scharlau [121] or Lang [90] for the definition and construction of the Grothendieck group and ring.) In particular, every element in Ŵ(F) is a difference of two isometry classes of nondegenerate symmetric bilinear forms over F. If 𝔟 is a nondegenerate symmetric bilinear form over F, we shall also write 𝔟 for the class in Ŵ(F). Thus, if α ∈ Ŵ(F), there exist nondegenerate symmetric bilinear forms 𝔟_{1} and 𝔟_{2} over F such that α = 𝔟_{1} − 𝔟_{2} in Ŵ(F). By definition, we have

𝔟_{1} − 𝔟_{2} = 𝔟′_{1} − 𝔟′_{2} in Ŵ(F)

if and only if there exists a nondegenerate symmetric bilinear form 𝔟′′ over F such that

(2.2)  𝔟_{1} ⊥ 𝔟′_{2} ⊥ 𝔟′′ ≃ 𝔟′_{1} ⊥ 𝔟_{2} ⊥ 𝔟′′.

As any hyperbolic form ℍ_{1}(V) is isometric to (dim V)ℍ_{1} over F, the ideal consisting of the hyperbolic forms over F in Ŵ(F) is the principal ideal ℍ_{1} by Lemma 2.1(3). The quotient W (F) := Ŵ(F)/(ℍ_{1}) is called the Witt ring of nondegenerate symmetric bilinear forms over F. Elements in W (F) are called Witt classes. Abusing notation, we shall also write 𝔟 ∈ W (F) for the Witt class of 𝔟 and often call it just the class of 𝔟. The operations in W (F) (and Ŵ(F)) shall be denoted by + and ·.

By (1.16), we have

⟨a, −a⟩ = 0 in W (F)

for all a ∈ F^{×} and in all characteristics. In particular, ⟨−1⟩ = −⟨1⟩ = −1 in W (F), hence the additive inverse of the Witt class of any nondegenerate symmetric bilinear form 𝔟 in W (F) is represented by the form −𝔟. It follows that if α ∈ W (F) then there exists a nondegenerate bilinear form 𝔟 such that α = 𝔟 in W (F).

Exercise 2.3 (cf. Scharlau [121], p. 22). Let 𝔟 be a metabolic symmetric bilinear form and V a lagrangian of 𝔟. Show that

𝔟 ⊥ (−𝔟) ≃ ℍ(V) ⊥ (−𝔟).

In particular, 𝔟 = ℍ(V) in Ŵ(F). Use this to give another proof that 𝔠 + (−𝔠) = 0 in W (F) for every nondegenerate form 𝔠.

The Witt Cancellation Theorem 1.28 allows us to conclude the following.

Proposition 2.4. Let 𝔟_{1} and 𝔟_{2} be anisotropic symmetric bilinear forms. Then the following are equivalent:

(1) 𝔟_{1} ≃ 𝔟_{2}.

(2) 𝔟_{1} = 𝔟_{2} in Ŵ(F).

(3) 𝔟_{1} = 𝔟_{2} in W (F).

Proof. The implications (1) ⇒ (2) ⇒ (3) are easy.

(3) ⇒ (1): By definition of the Witt ring, 𝔟_{1} + nℍ = 𝔟_{2} + mℍ in Ŵ(F) for some n, m ≥ 0. It follows from the definition of the Grothendieck-Witt ring that

𝔟_{1} ⊥ nℍ ⊥ 𝔟 ≃ 𝔟_{2} ⊥ mℍ ⊥ 𝔟

for some nondegenerate form 𝔟. Thus 𝔟_{1} ⊥ nℍ ⊥ 𝔟 ⊥ (−𝔟) ≃ 𝔟_{2} ⊥ mℍ ⊥ 𝔟 ⊥ (−𝔟) and 𝔟_{1} ≃ 𝔟_{2} by Corollary 1.28. □

We also have:

Corollary 2.5. 𝔟 = 0 in W (F) if and only if 𝔟 is metabolic.

It follows from Proposition 2.4 that every Witt class in W (F) contains (up to isometry) a unique anisotropic form. As every anisotropic bilinear form is diagonalizable by Corollary 1.19, we have a ring epimorphism

(2.6)  ℤ[F^{×}/F^{×2}] → W (F) given by ∑_{i} n_{i}(a_{i}F^{×2}) ↦ ∑_{i} n_{i}⟨a_{i}⟩.

Proposition 2.7. A homomorphism of fields F → K induces ring homomorphisms

r_{K/F} : Ŵ(F) → Ŵ(K) and r_{K/F} : W (F) → W (K).

If K/F is purely transcendental, then these maps are injective.

Proof. Let 𝔟 be symmetric bilinear form over F. Define r_{K/F} (𝔟) on K ⊗_{F} V_{𝔟} by

r_{K/F} (𝔟)(x ⊗ v, y ⊗ w) = xy𝔟(v, w)

for all x, y ∈ K and for all v, w ∈ V_{𝔟}. This construction is compatible with orthogonal sums and tensor products of symmetric bilinear forms.

As r_{K/F} (𝔟) is hyperbolic if 𝔟 is, it follows that 𝔟 ↦ r_{K/F} (𝔟) induces the desired maps. These are ring homomorphisms.

The last statement follows by Lemma 1.21. □

If K/F is a field extension, the ring homomorphisms r_{K/F} defined above are called restriction homomorphisms. Of course, the maps r_{K/F} are the unique homomorphisms such that r_{K/F} (𝔟) = 𝔟_{K}.

3. Chain equivalence

Two nondegenerate diagonal symmetric bilinear forms 𝔞 = ⟨a_{1}, a_{2}, …, a_{n}⟩ and 𝔟 = ⟨b_{1}, b_{2}, …, b_{n}⟩, are called simply chain equivalent if either n = 1 and a_{1}F^{×2} = b_{1}F^{×2} or n ≥ 2 and ⟨a_{i}, a_{j}⟩ ≃ ⟨b_{i}, b_{j}⟩ for some indices i ≠ j and a_{k} = b_{k} for every k ≠ i, j. Two nondegenerate diagonal forms 𝔞 and 𝔟 are called chain equivalent (we write 𝔞 ≈ 𝔟) if there is a chain of forms 𝔟_{1} = 𝔞, 𝔟_{2}, …, 𝔟_{m} = 𝔟 such that 𝔟_{i} and 𝔟_{i+1} are simply chain equivalent for all i ∈ [1, m − 1]. Clearly, 𝔞 ≈ 𝔟 implies 𝔞 ≃ 𝔟.

Note that as the symmetric group S_{n} is generated by transpositions, we have

⟨a_{1}, a_{2}, …, a_{n}⟩ ≈ ⟨a_{σ(1)}, a_{σ(2)}, …, a_{σ(n)}⟩

for every σ ∈ S_{n}.

Lemma 3.1. Every nondegenerate diagonal form is chain equivalent to an orthogonal sum of an anisotropic diagonal form and metabolic binary diagonal forms ⟨a, −a⟩, a ∈ F^{×}.

Proof. By induction, it is sufficient to prove that any isotropic diagonal form 𝔟 is chain equivalent to ⟨a, −a⟩ ⊥ 𝔟′ for some diagonal form 𝔟′ and a ∈ F^{×}. Let {v_{1}, …, v_{n}} be the orthogonal basis of 𝔟 and set 𝔟(v_{i}, v_{i}) = a_{i}. Choose an isotropic vector v with the smallest number k of nonzero coordinates. Changing the order of the v_{i} if necessary, we may assume that v = ∑_{i=1}^{k} c_{i}v_{i} for nonzero c_{i} ∈ F and k ≥ 2. We prove the statement by induction on k. If k = 2, the restriction of 𝔟 to the plane F v_{1} ⊕ F v_{2} is metabolic and therefore is isomorphic to ⟨a, −a⟩ for some a ∈ F^{×} by Example 1.22(3), hence 𝔟 ≈ ⟨a, −a⟩ ⊥ ⟨a_{3}, …, a_{n}⟩.

If k > 2, the vector v′_{1} = c_{1}v_{1}+c_{2}v_{2} is anisotropic. Complete v′_{1} to an orthogonal basis {v′_{1}, v′_{2}} of F v_{1} ⊕ F v_{2} by Corollary 1.7 and set a′_{i} = 𝔟(v′_{i}, v′_{i}), i = 1, 2. Then ⟨a_{1}, a_{2}⟩ ≃ ⟨a′_{1}, a′_{2}⟩ and 𝔟 ≈ ⟨a′_{1}, a′_{2}, a_{3}, …, a_{n}⟩. The vector v has k − 1 nonzero coordinates in the orthogonal basis {v′_{1}, v′_{2}, v_{3}, …, v_{n}}. Applying the induction hypothesis to the diagonal form ⟨a′_{1}, a′_{2}, a_{3}, …, a_{n}⟩ completes the proof. □

Lemma 3.2 (Witt Chain Equivalence). Two anisotropic diagonal forms are chain equivalent if and only if they are isometric.

Proof. Let {v_{1}, …, v_{n}} and {u_{1}, …, u_{n}} be two orthogonal bases of the bilinear form 𝔟 with 𝔟(v_{i}, v_{i}) = a_{i} and 𝔟(u_{i}, u_{i}) = b_{i}. We must show that ⟨a_{1}, …, a_{n}⟩ ≈ ⟨b_{1}, …, b_{n}⟩. We do this by double induction on n and the number k of nonzero coefficients of u_{1} in the basis {v_{i}}. Changing the order of the v_{i} if necessary, we may assume that u_{1} = ∑_{i=1}^{k} c_{i}v_{i} for some nonzero c_{i} ∈ F.

If k = 1, i.e., u_{1} = c_{1}v_{1}, the two (n − 1)-dimensional subspaces generated by the v_{i}’s and u_{i}’s respectively, with i ≥ 2, coincide. By the induction hypothesis and Witt Cancellation (Corollary 1.28), ⟨a_{2}, …, a_{n}⟩ ≈ ⟨b_{2}, …, b_{n}⟩, hence ⟨a_{1}, a_{2}, …, a_{n}⟩ ≈ ⟨a_{1}, b_{2}, …, b_{n}⟩ ≈ ⟨b_{1}, b_{2}, …, b_{n}⟩.

If k ≥ 2, set v′_{1} = c_{1}v_{1} + c_{2}v_{2}. As 𝔟 is anisotropic, a′_{1} = 𝔟(v′_{1}, v′_{1}) is nonzero. Choose an orthogonal basis {v′_{1}, v′_{2}} of F v_{1} ⊕ F v_{2} and set a′_{2} = 𝔟(v′_{2}, v′_{2}). We have ⟨a_{1}, a_{2}⟩ ≃ ⟨a′_{1}, a′_{2}⟩. The vector u_{1} has k − 1 nonzero coordinates in the basis {v′_{1}, v′_{2}, v_{3}, …, v_{n}}. By the induction hypothesis

⟨a_{1}, a_{2}, a_{3}, …, a_{n}⟩ ≈ ⟨a′_{1}, a′_{2}, a_{3}, …, a_{n}⟩ ≈ ⟨b_{1}, b_{2}, b_{3}, …, b_{n}⟩.  □

Exercise 3.3. Prove that a diagonalizable metabolic form 𝔟 is isometric to ⟨1, −1⟩⊗ 𝔟′ for some diagonalizable bilinear form 𝔟′.

4. Structure of the Witt ring

In this section, we give a presentation of the Witt-Grothendieck and Witt rings. The classes of even-dimensional anisotropic symmetric bilinear forms generate an ideal I(F) in the Witt ring. We also derive a presentation for it and its square, I(F)^{2}.

4.A. The presentation of Ŵ(F) and W (F). We turn to determining presentations of Ŵ(F) and W (F). The generators will be the isometry classes of nondegenerate 1-dimensional symmetric bilinear forms. The defining relations arise from the following:

Lemma 4.1. Let a, b ∈ F^{×} and z ∈ D(⟨a, b⟩). Then ⟨a, b⟩ ≃ ⟨z, abz⟩. In particular, if a + b ≠ 0, then

(4.2)  ⟨a, b⟩ ≃ ⟨a + b, ab(a + b)⟩.

Proof. By Corollary 1.7, we have ⟨a, b⟩ ≃ ⟨z, d⟩ for some d ∈ F^{×}. Comparing determinants, we must have abF^{×2} = dzF^{×2} so dF^{×2} = abzF^{×2}. □

The isometry (4.2) is often called the Witt relation.

Define an abelian group W′(F) by generators and relations. Generators are isometry classes of nondegenerate 1-dimensional symmetric bilinear forms. For any a ∈ F^{×} we write [a] for the generator — the isometry class of the form ⟨a⟩. Note that [ax^{2}] = [a] for every a, x ∈ F^{×}. The relations are

(4.3)  [a] + [b] = [a + b] + [ab(a + b)]

for all a, b ∈ F^{×} such that a + b ≠ 0.

Lemma 4.4. If ⟨a, b⟩ ≃ ⟨c, d⟩, then [a] + [b] = [c] + [d] in W′(F).

Proof. As ⟨a, b⟩ ≃ ⟨c, d⟩, we have abF^{×2} = det⟨a, b⟩ = det⟨c, d⟩ = cdF^{×2} and d = abcz^{2} for some z ∈ F^{×}. Since c ∈ D(⟨a, b⟩), there exist x, y ∈ F satisfying c = ax^{2} + by^{2}. If x = 0 or y = 0, the statement is obvious, so we may assume that x, y ∈ F^{×}. It follows from (4.3) that

[a] + [b] = [ax^{2}] + [by^{2}] = [c] + [ax^{2}by^{2}c] = [c] + [d].  □

Lemma 4.5. We have [a] + [−a] = [b] + [−b] in W′(F) for all a, b ∈ F^{×}.

Proof. We may assume that a + b ≠ 0. From (4.3), we have

[−a] + [a + b] = [b] + [−ab(a + b)], [−b] + [a + b] = [a] + [−ab(a + b)].

The result follows. □

If char F ≠ 2, the forms ⟨a, −a⟩ and ⟨b, −b⟩ are isometric by Remark 1.15(2). Therefore, in this case Lemma 4.5 follows from Lemma 4.4.

Lemma 4.6. If ⟨a_{1}, …, a_{n}⟩ ≈ ⟨b_{1}, …, b_{n}⟩, then [a_{1}] + ⋯ + [a_{n}] = [b_{1}] + ⋯ + [b_{n}] in W′(F).

Proof. We may assume that the forms are simply chain equivalent. In this case the statement follows from Lemma 4.4. □

Theorem 4.7. The Grothendieck-Witt group Ŵ(F) is generated by the isometry classes of 1-dimensional symmetric bilinear forms that are subject to the defining relations ⟨a⟩ + ⟨b⟩ = ⟨a + b⟩ + ⟨ab(a + b)⟩ for all a, b ∈ F^{×} such that a + b ≠ 0.

Proof. It suffices to prove that the homomorphism W′(F) → Ŵ(F) taking [a] to ⟨a⟩ is an isomorphism. As 𝔟 ⊥ ⟨1⟩ is diagonalizable for any nondegenerate symmetric bilinear form 𝔟 by Corollary 1.18, the map is surjective. An element in the kernel is given by the difference of two diagonal forms 𝔟 = ⟨a_{1}, …, a_{n}⟩ and 𝔟′ = ⟨a′_{1}, …, a′_{n}⟩ such that 𝔟 = 𝔟′ in Ŵ(F). By the definition of Ŵ(F) and Corollary 1.18, there is a diagonal form 𝔟′′ such that 𝔟 ⊥ 𝔟′′ ≃ 𝔟′ ⊥ 𝔟′′. Replacing 𝔟 and 𝔟′ by 𝔟 ⊥ 𝔟′′ and 𝔟′ ⊥ 𝔟′′ respectively, we may assume that 𝔟 ≃ 𝔟′. It follows from Lemma 3.1 that 𝔟 ≈ 𝔟_{1} ⊥ 𝔟_{2} and 𝔟′ ≈ 𝔟′_{1} ⊥ 𝔟′_{2}, where 𝔟_{1}, 𝔟′_{1} are anisotropic diagonal forms and 𝔟_{2}, 𝔟′_{2} are orthogonal sums of metabolic planes ⟨a, −a⟩ for various a ∈ F^{×}. It follows from the Corollary 1.28 that 𝔟_{1} ≃ 𝔟′_{1}, and therefore, 𝔟_{1} ≈ 𝔟′_{1} by Lemma 3.2. Note that the dimension of 𝔟_{2} and 𝔟′_{2} are equal. By Lemmas 4.5 and 4.6, we conclude that [a_{1}] + ⋯ + [a_{n}] = [a′_{1}] + ⋯ + [a′_{n}] in W′(F). □

Since the Witt class in W (F) of the hyperbolic plane ℍ_{1} is equal to ⟨1, −1⟩ by Remark 1.15(4), Theorem 4.7 yields:

Theorem 4.8. The Witt group W (F) is generated by the isometry classes of 1dimensional symmetric bilinear forms that are subject to the following defining relations:

(1) ⟨1⟩ + ⟨−1⟩ = 0.

(2) ⟨a⟩ + ⟨b⟩ = ⟨a + b⟩ + ⟨ab(a + b)⟩ for all a, b ∈ F^{×} such that a + b ≠ 0.

If char F ≠ 2, the above is the well-known presentation of the Witt-Grothen-dieck and Witt groups first demonstrated by Witt in [139].

4.B. The presentation of I(F). The Witt-Grothendieck and Witt rings have a natural filtration that we now describe. Define the dimension map

dim : Ŵ(F) → ℤ given by dim x = dim 𝔟_{1} − dim 𝔟_{2} if x = 𝔟_{1} − 𝔟_{2}.

This is a well-defined map by (2.2).

We let Î(F) denote the kernel of this map. As

⟨a⟩ − ⟨b⟩ = (⟨1⟩ − ⟨b⟩) − (⟨1⟩ − ⟨a⟩) in Ŵ(F)

for all a, b ∈ F^{×}, the elements ⟨1⟩ − ⟨a⟩ with a ∈ F^{×} generate Î(F) as an abelian group.

It follows that Ŵ(F) is generated by the elements ⟨1⟩ and ⟨1⟩−⟨x⟩ with x ∈ F^{×}. Let I(F) denote the image of Î(F) in W (F). If a ∈ F^{×}, write ⟨⟨a⟩⟩_{b} or simply ⟨⟨a⟩⟩ for the binary symmetric bilinear form ⟨1, −a⟩_{b}. As Î(F) ∩ (ℍ_{1}) = 0, we have I(F) ≃ Î(F)/(Î(F) ∩ (ℍ_{1})) ≃ Î(F). Then the map Ŵ(F) → W (F) induces an isomorphism

Î(F) → I(F) given by ⟨1⟩ − ⟨x⟩ ↦ ⟨⟨x⟩⟩.

In particular, I(F) is the ideal in W (F) consisting of the Witt classes of evendimensional forms. It is called the fundamental ideal of W (F) and is generated by the classes ⟨⟨a⟩⟩ with a ∈ F^{×}. Note that if F → K is a homomorphism of fields, then r_{K/F} (Î(F)) ⊂ Î(K) and r_{K/F} (I(F)) ⊂ I(K).

The relations in Theorem 4.8 can be rewritten as

⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ = ⟨⟨a + b⟩⟩ + ⟨⟨ab(a + b)⟩⟩

for a, b ∈ F^{×} with a + b ≠ 0. We conclude:

Corollary 4.9. The group I(F) is generated by the isometry classes of dimension 2 symmetric bilinear forms ⟨⟨a⟩⟩ with a ∈ F^{×} subject to the defining relations:

(1) ⟨⟨1⟩⟩ = 0.

(2) ⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ = ⟨⟨a + b⟩⟩ + ⟨⟨ab(a + b)⟩⟩ for all a, b ∈ F^{×} such that a + b ≠ 0.

Let Î^{n}(F) := Î(F)^{n} be the nth power of Î(F). Then Î^{n}(F) maps isomorphically onto I^{n}(F) := I(F)^{n}, the nth power of I(F) in W (F). It defines the filtration

W (F) ⊃ I(F) ⊃ I^{2}(F) ⊃ ⋯ ⊃ I^{n}(F) ⊃ ⋯

in which we shall be interested.

For convenience, we let Î^{0}(F) = Ŵ(F) and I^{0}(F) = W (F).

We denote the tensor product ⟨⟨a_{1}⟩⟩ ⊗ ⟨⟨a_{2}⟩⟩ ⊗ ⋯ ⊗ ⟨⟨a_{n}⟩⟩ by
⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩_{b} or simply by ⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩

and call a form isometric to such a tensor product a bilinear n-fold Pfister form. (We call any form isometric to ⟨1⟩ a 0-fold Pfister form.) For n ≥ 1, the isometry classes of bilinear n-fold Pfister forms generate I^{n}(F) as an abelian group.

We shall be interested in relations between isometry classes of Pfister forms in W (F). We begin with a study of 1- and 2-fold Pfister forms.

Example 4.10. We have ⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ = ⟨⟨ab⟩⟩ + ⟨⟨a, b⟩⟩ in W (F). In particular, ⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ ≡ ⟨⟨ab⟩⟩ mod I^{2}(F).

As the hyperbolic plane is 2-dimensional, the dimension invariant induces a map

e_{0} : W (F) → ℤ/2ℤ defined by e_{0}(𝔟) = dim 𝔟 mod 2.

Clearly, this is a homomorphism with kernel the fundamental ideal I(F), so it induces an isomorphism

(4.11)  ē_{0} : W (F)/I(F) → ℤ/2ℤ.

By Corollary 1.24, we have a map

e_{1} : I(F) → F^{×}/F^{×2} defined by e_{1}(𝔟) = (−1)^{\frac{dim 𝔟}{2}} det 𝔟

called the signed determinant.

The map e_{1} is a homomorphism as det(𝔟 ⊥ 𝔟′) = det 𝔟 · det 𝔟′ and surjective as e_{1}(⟨⟨a⟩⟩) = aF^{×2}. Clearly, e_{1}(⟨⟨a, b⟩⟩) = F^{×2}, so e_{1} induces an epimorphism

(4.12)  ē_{1} : I(F)/I^{2}(F) → F^{×}/F^{×2}.

We have

Proposition 4.13. The kernel of e_{1} is I^{2}(F) and the map ē_{1} : I(F)/I^{2}(F) → F^{×}/F^{×2} is an isomorphism.

Proof. Let f_{1} : F^{×}/F^{×2} → I(F)/I^{2}(F) be given by aF^{×2} ↦ ⟨⟨a⟩⟩ + I^{2}(F). This is a homomorphism by Example 4.10 inverse to ē_{1}, since I(F) is generated by ⟨⟨a⟩⟩, a ∈ F^{×}. □

For fields of characteristic different than 2 this is Pfister’s characterization of I^{2}(F) (cf. [110, Satz 13, Kor.]).

4.C. The presentation of I^{2}(F). We turn to I^{2}(F).

Lemma 4.14. Let a, b ∈ F^{×}. Then ⟨⟨a, b⟩⟩ = 0 in W (F) if and only if either a ∈ F^{×2} or b ∈ D(⟨⟨a⟩⟩). In particular, ⟨⟨a, 1 − a⟩⟩ = 0 in W (F) for any a ≠ 1 in F^{×}.

Proof. Suppose that ⟨⟨a⟩⟩ is anisotropic. Then ⟨⟨a, b⟩⟩ = 0 in W (F) if and only if b⟨⟨a⟩⟩ ≃ ⟨⟨a⟩⟩ by Proposition 2.4 if and only if b ∈ G(⟨⟨a⟩⟩) = D(⟨⟨a⟩⟩) by Example 1.14. □

Isometries of bilinear 2-fold Pfister forms are easily established using isometries of binary forms. For example, we have

Lemma 4.15. Let a, b ∈ F^{×} and x, y ∈ F. Let z = ax^{2} + by^{2} ≠ 0. Then:

(1) ⟨⟨a, b⟩⟩ ≃ ⟨⟨a, b(y^{2} − ax^{2})⟩⟩ if y^{2} − ax^{2} ≠ 0.

(2) ⟨⟨a, b⟩⟩ ≃ ⟨⟨z, −ab⟩⟩.

(3) ⟨⟨a, b⟩⟩ ≃ ⟨⟨z, abz⟩⟩.

(4) If z is a square in F, then ⟨⟨a, b⟩⟩ is metabolic. In particular, if char F ≠ 2, then ⟨⟨a, b⟩⟩ ≃ 2ℍ_{1}.

Proof. (1): Let w = y^{2} − ax^{2}. We have

⟨⟨a, b⟩⟩ ≃ ⟨1, −a, −b, ab⟩ ≃ ⟨1, −a, −by^{2}, abx^{2}⟩ ≃ ⟨1, −a, −bw, abw⟩ ≃ ⟨⟨a, bw⟩⟩.

(2): We have

⟨⟨a, b⟩⟩ ≃ ⟨1, −a, −b, ab⟩ ≃ ⟨1, −ax^{2}, −by^{2}, ab⟩ ≃ ⟨1, −z, −zab, ab⟩ ≃ ⟨⟨z, −ab⟩⟩.

(3) follows from (1)and (2) while (4) follows from (2) and Remark 1.15(2). □

Explicit examples of such isometries are:

Example 4.16. Let a, b ∈ F^{×}, then:

(1) ⟨⟨a, 1⟩⟩ is metabolic.

(2) ⟨⟨a, −a⟩⟩ is metabolic.

(3) ⟨⟨a, a⟩⟩ ≃ ⟨⟨a, −1⟩⟩.

(4) ⟨⟨a, b⟩⟩ + ⟨⟨a, −b⟩⟩ = ⟨⟨a, −1⟩⟩ in W (F).

We turn to a presentation of I^{2}(F) first done for fields of characteristic not 2 in [30] and rediscovered by Suslin (cf. [129]). It is different from that for I(F) as we need a new generating relation. Indeed, the analogue of the Witt relation will be a consequence of our new relation and a metabolic relation.

Let \underline{I}_{2}(F) be the abelian group generated by all the isometry classes [𝔟] of bilinear 2-fold Pfister forms 𝔟 subject to the generating relations:

(1) [⟨⟨1, 1⟩⟩] = 0.

(2) [⟨⟨ab, c⟩⟩] + [⟨⟨a, b⟩⟩] = [⟨⟨a, bc⟩⟩] + [⟨⟨b, c⟩⟩] for all a, b, c ∈ F^{×}.

We call the second relation the cocycle relation.

Remark 4.17. The cocycle relation holds in I^{2}(F): Let a, b, c ∈ F^{×}. Then

⟨⟨ab, c⟩⟩ + ⟨⟨a, b⟩⟩ = ⟨1, −ab, −c, abc⟩ + ⟨1, −a, −b, ab⟩
= ⟨1, 1, −c, abc, −a, −b⟩
= ⟨1, −a, −bc, abc⟩ + ⟨1, −b, −c, bc⟩
= ⟨⟨a, bc⟩⟩ + ⟨⟨b, c⟩⟩

in I^{2}(F).

We begin by showing that the analogue of the Witt relation is a consequence of the other two relations.

Lemma 4.18. The relations (1) [⟨⟨a, 1⟩⟩] = 0 ,

(2) [⟨⟨a, c⟩⟩] + [⟨⟨b, c⟩⟩] = [⟨⟨(a + b), c⟩⟩] + [⟨⟨(a + b)ab, c⟩⟩]

hold in \underline{I}_{2}(F) for all a, b, c ∈ F^{×} if a + b ≠ 0.

Proof. Applying the cocycle relation to a, a, 1 shows that

[⟨⟨1, 1⟩⟩] + [⟨⟨a, a⟩⟩] = [⟨⟨a, a⟩⟩] + [⟨⟨a, 1⟩⟩].

The first relation now follows. Applying Lemma 4.15 and the cocycle relation to a, c, c shows that

(4.19)  \begin{matrix}[⟨⟨−a, c⟩⟩] + [⟨⟨a, c⟩⟩] = [⟨⟨ac, c⟩⟩] + [⟨⟨a, c⟩⟩] \\ = [⟨⟨−a, c⟩⟩] + [⟨⟨a, c⟩⟩] = [⟨⟨−1, c⟩⟩] \end{matrix}

for all c ∈ F^{×}.

Applying the cocycle relation to a(a + b), a, c yields

(4.20)  [⟨⟨a + b, c⟩⟩] + [⟨⟨a(a + b), a⟩⟩] = [⟨⟨a(a + b), ac⟩⟩] + [⟨⟨a, c⟩⟩]

and to a(a + b), b, c yields

(4.21)  [⟨⟨ab(a + b), c⟩⟩] + [⟨⟨a(a + b), b⟩⟩] = [⟨⟨a(a + b), bc⟩⟩] + [⟨⟨b, c⟩⟩].

Adding the equations (4.20) and (4.21) and then using the isometries

⟨⟨a(a + b), a⟩⟩ ≃ ⟨⟨a(a + b), −b⟩⟩ and ⟨⟨a(a + b), ac⟩⟩ ≃ ⟨⟨a(a + b), −bc⟩⟩

derived from Lemma 4.15, followed by using equation (4.19) yields

[⟨⟨a, c⟩⟩] + [⟨⟨b, c⟩⟩] − [⟨⟨(a + b), c⟩⟩] − [⟨⟨a + b)ab, c⟩⟩]
= [⟨⟨a(a + b), a⟩⟩] + [⟨⟨a(a + b), b⟩⟩] − [⟨⟨a(a + b), ac⟩⟩] − [⟨⟨a(a + b), bc⟩⟩]

= [⟨⟨a(a + b), −b⟩⟩] + [⟨⟨a(a + b), b⟩⟩] − [⟨⟨a(a + b), −bc⟩⟩] − [⟨⟨a(a + b), bc⟩⟩]

= [⟨⟨a(a + b), −1⟩⟩] − [⟨⟨a(a + b), −1⟩⟩] = 0.  □

Theorem 4.22. The ideal I^{2}(F) is generated as an abelian group by the isometry classes ⟨⟨a, b⟩⟩ of bilinear 2-fold Pfister forms for all a, b ∈ F^{×} subject to the generating relations:

(1) ⟨⟨1, 1⟩⟩ = 0.

(2) ⟨⟨ab, c⟩⟩ + ⟨⟨a, b⟩⟩ = ⟨⟨a, bc⟩⟩ + ⟨⟨b, c⟩⟩ for all a, b, c ∈ F^{×}.

Proof. Clearly, we have well-defined homomorphisms

g : \underline{I}_{2}(F) → I^{2}(F) induced by [𝔟] ↦ 𝔟

and

j : \underline{I}_{2}(F) → I(F) induced by [⟨⟨a, b⟩⟩] ↦ ⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ − ⟨⟨ab⟩⟩,

the latter being the composition with the inclusion I^{2}(F) ⊂ I(F) using Example 4.10.

We show that the map g : \underline{I}_{2}(F) → I^{2}(F) is an isomorphism. Define
γ : F^{×}/F^{×2} × F^{×}/F^{×2} → \underline{I}_{2}(F) by (aF^{×2}, bF^{×2}) ↦ [⟨⟨a, b⟩⟩].

This is clearly well-defined. For convenience, write (a) for aF^{×2}. Using (2), we see that

γ((b), (c)) − γ((ab), (c)) + γ((a), (bc)) − γ((a), (b))
= [⟨⟨b, c⟩⟩] − [⟨⟨ab, c⟩⟩] + [⟨⟨a, bc⟩⟩] − [⟨⟨a, b⟩⟩] = 0,

so γ is a 2-cocycle. By Lemma 4.18, we have [⟨⟨1, a⟩⟩] = 0 in \underline{I}_{2}(F), so γ is a normalized 2-cocycle. The map γ defines an extension N = (F^{×}/F^{×2}) × \underline{I}_{2}(F) of \underline{I}_{2}(F) by F^{×}/F^{×2} with

((a), α) + ((b), β) = ((ab), α + β + [⟨⟨a, b⟩⟩]).

As γ is symmetric, N is abelian. Let

h : N → I(F) be defined by ((a), α) ↦ ⟨⟨a⟩⟩ + j(α).

We see that the map h is a homomorphism:

h(((a), α) + ((b), β)) = h((ab), α + β + [⟨⟨a, b⟩⟩])
= ⟨⟨ab⟩⟩ + j(α) + j(β) + j([⟨⟨a, b⟩⟩])
= ⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ + j(α) + j(β)
= h((a), α) + h((b), β).

Thus we have a commutative diagram

0 ⟶ \underline{I}_{2}(F) ⟶ N ⟶ F^{×}/F^{×2} ⟶ 0
g↓  ^{h}↓  ^{f_{1}}↓
0 ⟶ I^{2}(F) ⟶ I(F) ⟶ I(F)/I^{2}(F) ⟶ 0

where f_{1} is the isomorphism inverse of ē_{1} in Proposition 4.13.

Let

k : I(F) → N be induced by ⟨⟨a⟩⟩ ↦ ((a), 0).

Using Lemma 4.15 and Corollary 4.9, we see that k is well-defined as

((a), 0) + ((b), 0) = ((ab), [⟨⟨a, b⟩⟩]) = ((ab), [⟨⟨a + b, ab(a + b)⟩⟩])
= ((a + b), 0) + ((ab(a + b), 0)

if a + b ≠ 0. As

k(⟨⟨a, b⟩⟩) = k(⟨⟨a⟩⟩ + ⟨⟨b⟩⟩ − ⟨⟨ab⟩⟩) = ((a), 0) + ((b), 0) − ((ab), 0)

= ((ab), [⟨⟨a, b⟩⟩])−((ab), 0) = ((ab), 0)+(1, [⟨⟨, a, b⟩⟩])−((ab), 0) = (1, [⟨⟨, a, b⟩⟩]),

we have

(k ∘ h)((c), [⟨⟨a, b⟩⟩]) = k(⟨⟨c⟩⟩ + ⟨⟨a, b⟩⟩) = ((c), [⟨⟨a, b⟩⟩])).

Hence k ∘ h is the identity on N. As (h ∘ k)(⟨⟨a⟩⟩) = ⟨⟨a⟩⟩, the composition h ∘ k is the identity on I(F). Thus h is an isomorphism, hence so is g. □

5. The Stiefel-Whitney map

In this section, we investigate Stiefel-Whitney maps. In the case of fields of characteristic different from 2, this was first defined by Milnor. We shall use facts about Milnor K-theory (cf. §100). We write

k_{∗}(F) := ∐_{n≥0} k_{n}(F)

for the graded ring K_{∗}(F)/2K_{∗}(F). Abusing notation, if {a_{1}, …, a_{n}} is a symbol in K_{n}(F), we shall also write it for its coset {a_{1}, …, a_{n}} + 2K_{n}(F).

The associated graded ring

GW_{∗}(F) = ∐_{n≥0} I^{n}(F)/I^{n+1}(F)

of W (F) with respect to the fundamental ideal I(F) is called the graded Witt ring of symmetric bilinear forms. Note that since 2 · I^{n}(F) = ⟨1, 1⟩ · I^{n}(F) ⊂ I^{n+1}(F) we have 2 · GW_{∗}(F) = 0.

By Example 4.10, the map F^{×} → I(F)/I^{2}(F) defined by a ↦ ⟨⟨a⟩⟩ + I^{2}(F) is a homomorphism. By the definition of the Milnor ring and Lemma 4.14, this map gives rise to a graded ring homomorphism

(5.1)  f_{∗} : k_{∗}(F) → GW_{∗}(F)

taking the symbol {a_{1}, a_{2}, …, a_{n}} to ⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩ + I^{n+1}(F). Since the graded ring GW_{∗}(F) is generated by the degree one component I(F)/I^{2}(F), the map f_{∗} is surjective.

Note that the map f_{0} : k_{0}(F) → W (F)/I(F) is the inverse of the map ē_{0} and the map f_{1} : k_{1}(F) → I(F)/I^{2}(F) is the inverse of the map ē_{1} (cf. Proposition 4.13).

Lemma 5.2. Let ⟨⟨a, b⟩⟩ and ⟨⟨c, d⟩⟩ be isometric bilinear 2-fold Pfister forms. Then {a, b} = {c, d} in k_{2}(F).

Proof. If the form ⟨⟨a, b⟩⟩ is metabolic, then b ∈ D(⟨⟨a⟩⟩) or a ∈ F^{×2} by Lemma 4.14. In particular, if ⟨⟨a, b⟩⟩ is metabolic, then {a, b} = 0 in k_{2}(F). Therefore, we may assume that ⟨⟨a, b⟩⟩ is anisotropic. Using Witt Cancellation 1.28, we see that c = ax^{2} + by^{2} − abz^{2} for some x, y, z ∈ F. If c ∉ aF^{×2}, let w = y^{2} − az^{2} ≠ 0. Then ⟨⟨a, b⟩⟩ ≃ ⟨⟨a, bw⟩⟩ ≃ ⟨⟨c, −abw⟩⟩ by Lemma 4.15 and {a, b} = {a, bw} = {c, −abw} in k_{2}(F) by Lemma 100.3. Hence we may assume that a = c. By Witt Cancellation, ⟨−b, ab⟩ ≃ ⟨−d, ad⟩, so bd ∈ D(⟨⟨a⟩⟩), i.e., bd = x^{2} − ay^{2} in F for some x, y ∈ F. Thus {a, b} = {a, d} by Lemma 100.3. □

Proposition 5.3. The homomorphism

e_{2} : I^{2}(F) → k_{2}(F) given by ⟨⟨a, b⟩⟩ ↦ {a, b}

is a well-defined surjection with Ker(e_{2}) = I^{3}(F). Moreover, e_{2} induces an isomorphism

ē_{2} : I^{2}(F)/I^{3}(F) → k_{2}(F).

Proof. By Lemma 5.2 and the presentation of I^{2}(F) in Theorem 4.22, the map is well-defined. Since

⟨⟨a, b, c⟩⟩ = ⟨⟨a, c⟩⟩ + ⟨⟨b, c⟩⟩ − ⟨⟨ab, c⟩⟩,

we have I^{3}(F) ⊂ Ker(e_{2}). As ē_{2} and f_{2} are inverses of each other, the result follows. □

Let 𝔉(F) be the free abelian group on the set of isometry classes of nondegenerate 1-dimensional symmetric bilinear forms. Then we have a group homomorphism

w : 𝔉(F) → k_{∗}(F)[[t]]^{×} given by ⟨a⟩ ↦ 1 + {a}t.

If a, b ∈ F^{×} satisfy a + b ≠ 0, then by Lemma 100.3, we have

w(⟨a⟩ + ⟨b⟩) = (1 + {a}t)(1 + {b}t)
= 1 + ({a} + {b})t + {a, b}t^{2}
= 1 + {ab}t + {a, b}t^{2}
= 1 + {ab(a + b)^{2}}t + {a + b, ab(a + b)}t^{2}
= w(⟨a + b⟩ + ⟨ab(a + b)⟩).

In particular, w factors through the relation ⟨a⟩ + ⟨b⟩ = ⟨a + b⟩ + ⟨ab(a + b)⟩ for all a, b ∈ F^{×} satisfying a + b ≠ 0, hence induces a group homomorphism

(5.4)  w : Ŵ(F) → k_{∗}(F)[[t]]^{×}

by Theorem 4.7 called the total Stiefel-Whitney map. If 𝔟 is a nondegenerate symmetric bilinear form and α is its class in Ŵ(F) define the total Stiefel-Whitney class w(𝔟) of 𝔟 to be w(α).

Example 5.5. If 𝔟 is a metabolic plane, then 𝔟 = ⟨a⟩ + ⟨−a⟩ in Ŵ(F) for some a ∈ F^{×}. (Note the hyperbolic plane equals ⟨1⟩+⟨−1⟩ in Ŵ(F) by Example 1.15(4), so w(𝔟) = 1 + {−1}t as {a, −a} = 1 in k_{2}(F) for any a ∈ F^{×}.)

Lemma 5.6. Let α = (⟨1⟩ − ⟨a_{1}⟩) ⋯ (⟨1⟩ − ⟨a_{n}⟩) in Ŵ(F). Let m = 2^{n−1}. Then

w(α) = (1 + \{a_{1}, …, a_{n}, \underbrace{−1, …, −1}_{m−n} \}t^{m})^{−1}.

Proof. As

α = ∑_{ε} s_{ε}⟨a_{1}^{ε_{1}} ⋯ a_{n}^{ε_{n}}⟩,

where the sum runs over all ε = (ε_{1}, …, ε_{n}) ∈ {0, 1}^{n} and s_{ε} = (−1)^{∑_{i} ε_{i}}, we have

w(α) = ∏_{ε} (1 + ∑_{i} ε_{i}{a_{i}}t)^{s_{ε}}.

Let

h = h(t_{1}, …, t_{n}) = ∏_{ε} (1 + ε_{1}t_{1}t + ⋯ + ε_{n}t_{n}t)^{−s_{ε}}

in ((ℤ/2ℤ)[[t]])[[t_{1}, …, t_{n}]], the ring of power series over ℤ/2ℤ in variables

t, t_{1}, …, t_{n}.

Substituting zero for any t_{i} in h, yields one, so

h = 1 + t_{1} ⋯ t_{n}g(t_{1}, …, t_{n})t^{n} for some g ∈ ((ℤ/2ℤ)[[t]])[[t_{1}, …, t_{n}]].

As {a, a} = {a, −1}, we have

w(α)^{−1} = 1 + {a_{1}, …, a_{n}}g({a_{1}}, …, {a_{n}})t^{n}

= 1 + {a_{1}, …, a_{n}}g({−1}, …, {−1})t^{n}.

We have with s a variable,

1 + g(s, …, s)t^{n} = h(s, …, s) = ∏_{ε} (1 + ∑_{i≥1} ε_{i}st)^{−s_{ε}} = (1 + st)^{m} = 1 + s^{m}t^{m}

as ∑ ε_{i} = 1 in ℤ/2ℤ exactly m times, so g(s, …, s) = (st)^{m−n} and the result follows. □

Let w_{0}(α) = 1 and

w(α) = 1 + ∑_{i≥1} w_{i}(α)t^{i}

for α ∈ Ŵ(F). The map w_{i} : Ŵ(F) → k_{i}(F) is called the ith Stiefel-Whitney class. Let α, β ∈ Ŵ(F). As w(α + β) = w(α)w(β), for every n ≥ 0, we have the Whitney Sum Formula

(5.7)  w_{n}(α + β) = ∑_{i+j=n} w_{i}(α)w_{j}(β).

Remark 5.8. Let K/F be a field extension and α ∈ Ŵ(F). Then

res_{K/F} (w_{i}(α)) = w_{i}(α_{K}) in k_{i}(F) for all i.

Corollary 5.9. Let m = 2^{n−1}. Then w_{j}(Î^{n}(F)) = 0 for j ∈ [1, m − 1] and w_{m} : Î^{n}(F) → k_{m}(F) is a group homomorphism mapping (⟨1⟩−⟨a_{1}⟩) ⋯ (⟨1⟩−⟨a_{n}⟩)

to \{a_{1}, …, a_{n}, \underbrace{−1, …, −1}_{m−n} \}.

Proof. Let α = (⟨1⟩−⟨a_{1}⟩) ⋯ (⟨1⟩−⟨a_{n}⟩). By Lemma 5.6, we have w_{i}(α) = 0 for i ∈ [1, m − 1]. The result follows from the Whitney formula (5.7). □

Let j : Î(F) → I(F) be the isomorphism sending ⟨1⟩ − ⟨a⟩ ↦ ⟨⟨a⟩⟩. Let w̃_{m} be the composition

I^{n}(F) ⟶^{j^{−1}} Î^{n}(F) ⟶^{w_{m}|_{Î^{n}(F)}} k_{m}(F).

Corollary 5.9 shows that w̃_{i} = e_{i} for i = 1, 2. The map w̃_{m} : I^{n}(F) → k_{m}(F) is a group homomorphism with I^{n+1}(F) ⊂ Ker(w̃_{m}) so it induces a homomorphism

w̄_{m} : I^{n}(F)/I^{n+1}(F) → k_{m}(F).

We have w̄_{i} = ē_{i} for i = 1, 2. The composition w̄_{m} ∘ f_{n} is multiplication by \{\underbrace{−1, …, −1}_{m−n}\}. In particular, w̄_{1} and w̄_{2} are isomorphisms, i.e.,

(5.10)  I^{2}(F) = Ker(w̃_{1}) and I^{3}(F) = Ker(w̃_{2})

and

(5.11)  Î^{2}(F) = Ker(w_{1}|_{Î(F)}) and Î^{3}(F) = Ker(w_{2}|_{Î^{2}(F)}).

This gives another proof for Propositions 4.13 and 5.3.

Remark 5.12. Let char F ≠ 2 and h_{F}^{2} : k_{2}(F) → H^{2}(F) be the norm-residue homomorphism defined in §101. If 𝔟 is a nondegenerate symmetric bilinear form, then h_{2} ∘ w_{2}(𝔟) is the classical Hasse-Witt invariant of 𝔟. (Cf. [89], Definition V.3.17, [121], Definition 2.12.7.) More generally, the Stiefel-Whitney classes defined above are compatible with Stiefel-Whitney classes defined by Delzant w′_{i} in [27], i.e., h_{i} ∘ w_{i} = w′_{i} for all i ≥ 0.

Example 5.13. Suppose that K is a real-closed field. (Cf. §95.) Then k_{i}(K) = ℤ/2ℤ for all i ≥ 0 and Ŵ(K) = ℤ ⊕ ℤξ with ξ = ⟨−1⟩ and ξ^{2} = 1. The Stiefel-Whitney map w : Ŵ(F) → k_{∗}(K)[[t]]^{×} is then the map n + mξ ↦ (1 + t)^{m}. In particular, if 𝔟 is a nondegenerate form, then w(𝔟) determines the signature of 𝔟. Hence if 𝔟 and 𝔠 are two nondegenerate symmetric bilinear forms over K, we have 𝔟 ≃ 𝔠 if and only if dim 𝔟 = dim 𝔠 and w(𝔟) = w(𝔠).

It should be noted that if 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ that w(𝔟) is not equal to w(α) = w̃([𝔟]) where α = (⟨1⟩ − ⟨a_{1}⟩) ⋯ (⟨1⟩ − ⟨a_{n}⟩) in Ŵ(F) as the following exercise shows.

Exercise 5.14. Let m = 2^{n−1}. If 𝔟 is the bilinear n-fold Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩, then

w(𝔟) = 1 + (\{\underbrace{−1, …, −1}_{m} \} + \{a_{1}, …, a_{n}, \underbrace{−1, …, −1}_{m−n} \})t^{m}.

The following fundamental theorem was proved by Orlov-Vishik-Voevodsky [107] in the case that char F ≠ 2 and by Kato [78] in the case that char F = 2.

Fact 5.15. The map f_{∗} : k_{∗}(F) → GW_{∗}(F) is a ring isomorphism.

For i = 0, 1, 2, we have proven that f_{i} is an isomorphism in (4.11), Proposition 4.13 , and Proposition 5.3, respectively.

6. Bilinear Pfister forms

The isometry classes of tensor products of nondegenerate binary symmetric bilinear forms representing one are quite interesting. These forms, called Pfister forms, whose properties over fields of characteristic different from 2 were discovered by Pfister in [108] and were named after him in [32], generate a filtration of the Witt ring by the powers of its fundamental ideal I(F). Properties of these forms in the case of characteristic 2 were first studied by Baeza in [15]. In this section, we derive the main elementary properties of these forms.

By Example 1.14, a bilinear 1-fold Pfister form 𝔟 = ⟨⟨a⟩⟩, a ∈ F^{×}, is round, i.e., D(⟨⟨a⟩⟩) = G(⟨⟨a⟩⟩). Because of this, the next proposition shows that there are many round forms and, in particular, bilinear Pfister forms are round.

Proposition 6.1. Let 𝔟 be a round bilinear form and let a ∈ F^{×}. Then: (1) The form ⟨⟨a⟩⟩ ⊗ 𝔟 is also round.

(2) If ⟨⟨a⟩⟩ ⊗ 𝔟 is isotropic, then either 𝔟 is isotropic or a ∈ D(𝔟).

Proof. Set 𝔠 = ⟨⟨a⟩⟩ ⊗ 𝔟.

(1): Since 1 ∈ D(𝔟), it suffices to prove that D(𝔠) ⊂ G(𝔠). Let c be a nonzero value of 𝔠. Write c = x − ay for some x, y ∈ D̃(𝔟). If y = 0, we have c = x ∈ D(𝔟) = G(𝔟) ⊂ G(𝔠). Similarly, y ∈ G(𝔠) if x = 0, hence c = −ay ∈ G(𝔠) as −a ∈ G(⟨⟨a⟩⟩) ⊂ G(𝔠).

Now suppose that x and y are nonzero. Since 𝔟 is round, x, y ∈ G(𝔟) and, therefore,

𝔠 = 𝔟 ⊥ (−a𝔟) ≃ 𝔟 ⊥ (−ayx^{−1})𝔟 = ⟨⟨ayx^{−1}⟩⟩ ⊗ 𝔟.

By Example 1.14, we know that 1 − ayx^{−1} ∈ G(⟨⟨ayx^{−1}⟩⟩) ⊂ G(𝔠). Since x ∈ G(𝔟) ⊂ G(𝔠), we have c = (1 − ayx^{−1})x ∈ G(𝔠).

(2): Suppose that 𝔟 is anisotropic. Since 𝔠 = 𝔟 ⊥ (−a𝔟) is isotropic, there exist x, y ∈ D(𝔟) with x − ay = 0. Therefore a = xy^{−1} ∈ D(𝔟) as D(𝔟) is closed under multiplication. □

Corollary 6.2. Bilinear Pfister forms are round.

Proof. 0-fold Pfister forms are round. □

Corollary 6.3. A bilinear Pfister form is either anisotropic or metabolic.

Proof. Suppose that 𝔠 is an isotropic bilinear Pfister form. We show that 𝔠 is metabolic by induction on the dimension of the 𝔠. We may assume that 𝔠 = ⟨⟨a⟩⟩⊗𝔟 for a Pfister form 𝔟. If 𝔟 is metabolic, then so is 𝔠. By the induction hypothesis, we may assume that 𝔟 is anisotropic. By Proposition 6.1 and Corollary 6.2, a ∈ D(𝔟) = G(𝔟). Therefore a𝔟 ≃ 𝔟 hence the form 𝔠 ≃ 𝔟 ⊥ (−a𝔟) ≃ 𝔟 ⊥ (−𝔟) is metabolic. □

Remark 6.4. Note that the only metabolic 1-fold Pfister form is ⟨⟨1⟩⟩. If char F ≠ 2, there is only one metabolic bilinear n-fold Pfister form for all n ≥ 1, viz., the hyperbolic one. It is universal by Corollary 1.25. If char F = 2, then there may exist many metabolic n-fold Pfister forms for n > 1.

Example 6.5. If char F = 2, a bilinear Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩ is anisotropic if and only if a_{1}, …, a_{n} are 2-independent. Indeed, [F^{2}(a_{1}, …, a_{n}) : F^{2}] < 2^{n} if and only if ⟨⟨a_{1}, …, a_{n}⟩⟩ is isotropic.

Corollary 6.6. Let char F ≠ 2 and z ∈ F^{×}. Then 2^{n}⟨⟨z⟩⟩ = 0 in W (F) if and only if z ∈ D(2^{n}⟨1⟩).

Proof. If z ∈ D(2^{n}⟨1⟩), then the Pfister form 2^{n}⟨⟨z⟩⟩ is isotropic hence metabolic by Corollary 6.3.

Conversely, suppose that 2^{n}⟨⟨z⟩⟩ is metabolic. Then 2^{n}⟨1⟩ = 2^{n}⟨z⟩ in W (F). If 2^{n}⟨1⟩ is isotropic, it is universal as char F ≠ 2, so z ∈ D(2^{n}⟨1⟩). If 2^{n}⟨1⟩ is anisotropic, then 2^{n}⟨1⟩ ≃ 2^{n}⟨z⟩ by Proposition 2.4, so z ∈ G(2^{n}⟨1⟩) = D(2^{n}⟨1⟩) by Corollary 6.2. □

As additional corollaries, we have the following two theorems of Pfister (cf. [109]). The first generalizes the well-known 2-, 4-, and 8-square theorems arising from quadratic extensions, quaternion algebras, and Cayley algebras.

Corollary 6.7. D(2^{n}⟨1⟩) is a group for every nonnegative integer n.

The level of a field F is defined to be

s(F) := min {n | the element − 1 is a sum of n squares}

or infinity if no such integer exists.

Corollary 6.8. The level s(F) of a field F, if finite, is a power of two.

Proof. Suppose that s(F) is finite. Then 2^{n} ≤ s(F) < 2^{n+1} for some n. By Proposition 6.1(2), with 𝔟 = 2^{n}⟨1⟩ and a = −1, we have −1 ∈ D(𝔟). Hence s(F) = 2^{n}. □

In [109], Pfister also showed that there exist fields of level 2^{n} for all n ≥ 0. (Cf. Lemma 31.3 below.)

6.A. Chain p-equivalence of bilinear Pfister forms. Since the isometry classes of 2-fold Pfister forms are easy to deal with, we use them to study nfold Pfister forms. We follow the development in [32] which we extend to all characteristics. The case of characteristic 2 was also independently done by Arason and Baeza in [6].

Definition 6.9. Let a_{1}, …, a_{n}, b_{1}, …, b_{n} ∈ F^{×} with n ≥ 1. We say that the forms ⟨⟨a_{1}, …, a_{n}⟩⟩ and ⟨⟨b_{1}, …, b_{n}⟩⟩ are simply p-equivalent if n = 1 and a_{1}F^{×2} = b_{1}F^{×2} or n ≥ 2 and there exist i, j ∈ [1, n] such that

⟨⟨a_{i}, a_{j}⟩⟩ ≃ ⟨⟨b_{i}, b_{j}⟩⟩ with i ≠ j and a_{l} = b_{l} for all l ≠ i, j.

We say bilinear n-fold Pfister forms 𝔟 and 𝔠 are chain p-equivalent if there exist bilinear n-fold Pfister forms 𝔟_{0}, …, 𝔟_{m} for some m such that 𝔟 = 𝔟_{0}, 𝔠 = 𝔟_{m} and 𝔟_{i} is simply p-equivalent to 𝔟_{i+1} for each i ∈ [0, m − 1].

Chain p-equivalence is clearly an equivalence relation on the set of anisotropic bilinear forms of the type ⟨⟨a_{1}, …, a_{n}⟩⟩ with a_{1}, …, a_{n} ∈ F^{×} and is denoted by ≈. As transpositions generate the symmetric group, we have ⟨⟨a_{1}, …, a_{n}⟩⟩ ≈ ⟨⟨a_{σ(1)}, …, a_{σ(n)}⟩⟩ for every permutation σ of {1, …, n}. We shall show the following result:

Theorem 6.10. Let ⟨⟨a_{1}, …, a_{n}⟩⟩ and ⟨⟨b_{1}, …, b_{n}⟩⟩ be anisotropic. Then

⟨⟨a_{1}, …, a_{n}⟩⟩ ≃ ⟨⟨b_{1}, …, b_{n}⟩⟩

if and only if

⟨⟨a_{1}, …, a_{n}⟩⟩ ≈ ⟨⟨b_{1}, …, b_{n}⟩⟩.

Of course, we need only show isometric anisotropic bilinear Pfister forms are chain p-equivalent. We shall do this in a number of steps. If 𝔟 is an n-fold Pfister form, then we can write 𝔟 ≃ 𝔟′ ⊥ ⟨1⟩. If 𝔟 is anisotropic, then 𝔟′ is unique up to isometry and we call it the pure subform of 𝔟.

Lemma 6.11. Suppose that 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ is anisotropic. Let −b ∈ D(𝔟′). Then there exist b_{2}, …, b_{n} ∈ F^{×} such that 𝔟 ≈ ⟨⟨b, b_{2}, …, b_{n}⟩⟩.

Proof. We induct on n, the case n = 1 being trivial. Let 𝔠 = ⟨⟨a_{1}, …, a_{n−1}⟩⟩ so 𝔟′ ≃ 𝔠′ ⊥ −a_{n}𝔠 by Witt Cancellation 1.28. Write

−b = −x + a_{n}y with − x ∈ D̃(𝔠′), −y ∈ D̃(𝔠).

If y = 0, then x ≠ 0 and we finish by induction, so we may assume that 0 ≠ y = y_{1} + z^{2} with −y_{1} ∈ D̃(𝔠′) and z ∈ F. If y_{1} ≠ 0, then 𝔠 ≈ ⟨⟨y_{1}, …, y_{n−1}⟩⟩ for some y_{i} ∈ F^{×} and, using Lemma 4.15, we get

(6.12) 𝔠 ≈ ⟨⟨y_{1}, …, y_{n−1}, a_{n}⟩⟩ ≈ ⟨⟨y_{1}, …, y_{n−1}, −a_{n}y⟩⟩ ≈ ⟨⟨a_{1}, …, a_{n−1}, −a_{n}y⟩⟩.

This is also true if y_{1} = 0. If x = 0, we are done. If not 𝔠 ≈ ⟨⟨x, x_{2}, …, x_{n−1}⟩⟩ for some x_{i} ∈ F^{×} and

𝔟 ≈ ⟨⟨x, x_{2}, …, x_{n−1}, −a_{n}y⟩⟩ ≈ ⟨⟨a_{n}xy, x_{2}, …, x_{n−1}, −a_{n}y + x⟩⟩
≈ ⟨⟨a_{n}xy, x_{2}, …, x_{n−1}, b⟩⟩

by Lemma 4.15(2) as needed. □

The argument to establish equation (6.12) yields:

Corollary 6.13. Let 𝔟 = ⟨⟨x_{1}, …, x_{n}⟩⟩ and y ∈ D(𝔟). Let z ∈ F^{×}. If 𝔟 ⊗ ⟨⟨z⟩⟩ is anisotropic, then ⟨⟨x_{1}, …, x_{n}, z⟩⟩ ≈ ⟨⟨x_{1}, …, x_{n}, yz⟩⟩.

We also have the following generalization of Lemma 4.14:

Corollary 6.14. Let 𝔟 be an anisotropic bilinear Pfister form over F and let a ∈ F^{×}. Then ⟨⟨a⟩⟩ · 𝔟 = 0 in W (F) if and only if either a ∈ F^{×2} or 𝔟 ≃ ⟨⟨b⟩⟩ ⊗ 𝔠 for some b ∈ D(⟨⟨a⟩⟩) and bilinear Pfister form 𝔠. In the latter case, ⟨⟨a, b⟩⟩ is metabolic.

Proof. Clearly ⟨⟨a, b⟩⟩ = 0 in W (F) if b ∈ D(⟨⟨a⟩⟩). Conversely, suppose that ⟨⟨a⟩⟩ ⊗ 𝔟 = 0. Hence a ∈ G(𝔟) = D(𝔟) by Corollary 6.2. Write a = x^{2} − b for some x ∈ F and −b ∈ D̃(𝔟′). If b = 0, then a ∈ F^{×2}. Otherwise, b ∈ D(⟨⟨a⟩⟩) and 𝔟 ≃ ⟨⟨b⟩⟩ ⊗ 𝔠 for some bilinear Pfister form 𝔠 by Lemma 6.11. □

The following generalization of Lemma 6.11 is very useful in computation and is the key to proving further relations among Pfister forms.

Proposition 6.15. Let 𝔟 = ⟨⟨a_{1}, …, a_{m}⟩⟩ and 𝔠 = ⟨⟨b_{1}, …, b_{n}⟩⟩ be such that 𝔟 ⊗ 𝔠 is anisotropic. Let −c ∈ D(𝔟 ⊗ 𝔠′), then

⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n}⟩⟩ ≈ ⟨⟨a_{1}, …, a_{m}, c_{1}, c_{2}, …, c_{n−1}, c⟩⟩

for some c_{1}, …, c_{n−1} ∈ F^{×}.

Proof. We induct on n. If n = 1, then −c = yb_{1} for some −y ∈ D(𝔟) and this case follows by Corollary 6.13, so we assume that n > 1. Let 𝔡 = ⟨⟨b_{1}, …, b_{n−1}⟩⟩.

Then 𝔠′ ≃ b_{n}𝔡 ⊥ 𝔡′ so 𝔟𝔠′ ≃ b_{n}𝔟 ⊗ 𝔡 ⊥ 𝔟 ⊗ 𝔡′. Write 0 ≠ −c = b_{n}y − z with −y ∈ D̃(𝔟 ⊗ 𝔠) and −z ∈ D̃(𝔟 ⊗ 𝔠′). If z = 0, then x ≠ 0 and

⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n}⟩⟩ ≈ ⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n−1}, −yb_{n}⟩⟩

by Corollary 6.13 and we are done. So we may assume that z ≠ 0. By induction

⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n−1}⟩⟩ ≈ ⟨⟨a_{1}, …, a_{m}, c_{1}, c_{2}, …, c_{n−2}, z⟩⟩

for some c_{1}, …, c_{n−2} ∈ F^{×}. If y = 0, tensoring this by ⟨1, −b_{n}⟩ completes the proof, so we may assume that y ≠ 0. Then

⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n}⟩⟩ ≈ ⟨⟨a_{1}, …, a_{m}, b_{1}, …, b_{n−1}, −yb_{n}⟩⟩

≈ ⟨⟨a_{1}, …, a_{m}, c_{1}, …, c_{n−2}, z, −yb_{n}⟩⟩ ≈ ⟨⟨a_{1}, …, a_{m}, c_{1}, …, c_{n−2}, z − yb_{n}, zyb_{n}⟩⟩

≈ ⟨⟨a_{1}, …, a_{m}, c_{1}, …, c_{n−2}, c, zyb_{n}⟩⟩

by Lemma 4.15(2). This completes the proof. □

Corollary 6.16 (Common Slot Property). Let

⟨⟨a_{1}, …, a_{n−1}, x⟩⟩ and ⟨⟨b_{1}, …, b_{n−1}, y⟩⟩

be isometric anisotropic bilinear forms. Then there exists a z ∈ F^{×} satisfying

⟨⟨a_{1}, …, a_{n−1}, z⟩⟩ = ⟨⟨a_{1}, …, a_{n−1}, x⟩⟩ and ⟨⟨b_{1}, …, b_{n−1}, z⟩⟩ = ⟨⟨b_{1}, …, b_{n−1}, y⟩⟩.

Proof. Let 𝔟 = ⟨⟨a_{1}, …, a_{n−1}⟩⟩ and 𝔠 = ⟨⟨b_{1}, …, b_{n−1}⟩⟩. As x𝔟 − y𝔠 = 𝔟′ − 𝔠′ in W (F), the form x𝔟 ⊥ −y𝔠 is isotropic. Hence there exists a z ∈ D(x𝔟) ∩ D(y𝔠). The result follows by Proposition 6.15. □

A nondegenerate symmetric bilinear form 𝔟 is called a general bilinear n-fold Pfister form if 𝔟 ≃ a𝔠 for some a ∈ F^{×} and bilinear n-fold Pfister form 𝔠. As Pfister forms are round, a general Pfister form is a Pfister form if and only if it represents one.

Corollary 6.17. Let 𝔠 and 𝔟 be general anisotropic bilinear Pfister forms. If 𝔠 is a subform of 𝔟, then 𝔟 ≃ 𝔠 ⊗ 𝔡 for some bilinear Pfister form 𝔡.

Proof. If 𝔠 = c𝔠_{1} for some Pfister form 𝔠_{1} and c ∈ F^{×}, then 𝔠_{1} is a subform of c𝔟. In particular, c𝔟 represents one, so it is a Pfister form. Replacing 𝔟 by c𝔟 and 𝔠 by c𝔠, we may assume both are Pfister forms.

Let 𝔠 ≃ ⟨⟨a_{1}, …, a_{n}⟩⟩ with a_{i} ∈ F^{×}. By Witt Cancellation 1.28, we have 𝔠′ is isometric to a subform of 𝔟′, hence 𝔟 ≃ ⟨⟨a_{1}⟩⟩⊗𝔡_{1} for some Pfister form 𝔡_{1} by Lemma 6.11. By induction, there exists a Pfister form 𝔡_{k} satisfying 𝔟 ≃ ⟨⟨a_{1}, …, a_{k}⟩⟩ ⊗ 𝔡_{k}. By Witt Cancellation 1.28, we have ⟨⟨a_{1}, …, a_{k}⟩⟩ ⊗ ⟨⟨a_{k+1}, …, a_{n}⟩⟩′ is a subform of ⟨⟨a_{1}, …, a_{k}⟩⟩ ⊗ 𝔡′_{k} so −a_{k+1} ∈ D(⟨⟨a_{1}, …, a_{k}⟩⟩ ⊗ 𝔡′_{k}). By Proposition 6.15, we complete the induction step. □

Let 𝔟 and 𝔠 be general Pfister forms. We say that 𝔠 divides 𝔟 if 𝔟 ≃ 𝔠 ⊗ 𝔡 for some Pfister form 𝔡. The corollary says that 𝔠 divides 𝔟 if and only if it is isometric to a subform of 𝔟.

We now prove Theorem 6.10.

Proof. Let 𝔞 = ⟨⟨a_{1}, …, a_{n}⟩⟩ and 𝔟 = ⟨⟨b_{1}, …, b_{n}⟩⟩ be isometric over F. Clearly we may assume that n > 1. By Lemma 6.11, we have 𝔞 ≈ ⟨⟨b_{1}, a′_{2}, …, a′_{n}⟩⟩ for some a′_{i} ∈ F^{×}. Suppose that we have shown 𝔞 ≈ ⟨⟨b_{1}, …, b_{m}, a′_{m+1}, …, a′_{n}⟩⟩ for some m. By Witt Cancellation 1.28,

⟨⟨b_{1}, …, b_{m}⟩⟩ ⊗ ⟨⟨b_{m+1}, …, b_{n}⟩⟩′ ≃ ⟨⟨b_{1}, …, b_{m}⟩⟩ ⊗ ⟨⟨a′_{m+1}, …, a′_{n}⟩⟩′,

so −b_{m+1} ∈ D(⟨⟨b_{1}, …, b_{m}⟩⟩ ⊗ ⟨⟨a′_{m+1}, …, a′_{n}⟩⟩′). By Proposition 6.15, we have

𝔞 ≈ ⟨⟨b_{1}, …, b_{m+1}, a′′_{m+2}, …, a′′_{n}⟩⟩

for some a′′_{i} ∈ F^{×}. This completes the induction step. □

We need the following theorem of Arason and Pfister (cf. [10]):

Theorem 6.18 (Hauptsatz). Let 0 ≠ 𝔟 be an anisotropic form lying in I^{n}(F). Then dim 𝔟 ≥ 2^{n}.

We shall prove this theorem in Theorem 23.7 below. Using it we show:

Corollary 6.19. Let 𝔟 and 𝔠 be two anisotropic general bilinear n-fold Pfister forms. If 𝔟 ≡ 𝔠 mod I^{n+1}(F), then 𝔟 ≃ a𝔠 for some a ∈ F^{×}. In addition, if D(𝔟) ∩ D(𝔠) ≠ ∅, then 𝔟 ≃ 𝔠.

Proof. Choose a ∈ F^{×} such that 𝔟 ⊥ −a𝔠 is isotropic. By the Hauptsatz, this form must be metabolic. By Proposition 2.4, we have 𝔟 ≃ a𝔠.

Suppose that x ∈ D(𝔟) ∩ D(𝔠). Then 𝔟 ⊥ −𝔠 is isotropic and one can take a = 1. □

Theorem 6.20. Let a_{1}, …, a_{n}, b_{1}, …, b_{n} ∈ F^{×}. The following are equivalent: (1) ⟨⟨a_{1}, …, a_{n}⟩⟩ = ⟨⟨b_{1}, …, b_{n}⟩⟩ in W (F).

(2) ⟨⟨a_{1}, …, a_{n}⟩⟩ ≡ ⟨⟨b_{1}, …, b_{n}⟩⟩ mod I^{n+1}(F).
(3) {a_{1}, …, a_{n}} = {b_{1}, …, b_{n}} in K_{n}(F)/2K_{n}(F).

Proof. Let 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ and 𝔠 = ⟨⟨b_{1}, …, b_{n}⟩⟩. As metabolic Pfister forms are trivial in W (F) and any bilinear n-fold Pfister form lying in I^{n+1}(F) must be metabolic by the Hauptsatz 6.18, we may assume that 𝔟 and 𝔠 are both anisotropic.

(2) ⇒ (1) follows from Corollary 6.19.

(1) ⇒ (3): By Theorem 6.10, we have ⟨⟨a_{1}, …, a_{n}⟩⟩ ≈ ⟨⟨b_{1}, …, b_{n}⟩⟩, so it suffices to show that (3) holds if

⟨⟨a_{i}, a_{j}⟩⟩ ≃ ⟨⟨b_{i}, b_{j}⟩⟩ with i ≠ j and a_{l} = b_{l} for all l ≠ i, j.

As {a_{i}, a_{j}} = {b_{i}, b_{j}} by Proposition 5.3, statement (3) follows.

(3) ⇒ (2) follows from (5.1). □

Arason proved and used the Common Slot Property 6.16 in [4] to give an independent proof in the case of characteristic different from 2 of Theorem 6.20 which was first proven in [32].

6.B. Linkage of bilinear Pfister forms. We derive some further properties of bilinear Pfister forms that we shall need later.

Proposition 6.21. Let 𝔟_{1} and 𝔟_{2} be two anisotropic general bilinear Pfister forms. Let 𝔠 be a general r-fold Pfister form with r ≥ 0 that is isometric to a subform of 𝔟_{1} and to a subform of 𝔟_{2}. If 𝔦(𝔟_{1} ⊥ −𝔟_{2}) > 2^{r}, then there exists a k-fold Pfister form 𝔡 with k > 0 such that 𝔠 ⊗ 𝔡 is isometric to a subform of 𝔟_{1} and to a subform of 𝔟_{2}. Furthermore, 𝔦(𝔟_{1} ⊥ −𝔟_{2}) = 2^{r+k}.

Proof. By Corollary 6.17, there exist Pfister forms 𝔡_{1} and 𝔡_{2} such that 𝔟_{1} ≃ 𝔠 ⊗ 𝔡_{1} and 𝔟_{2} ≃ 𝔠 ⊗ 𝔡_{2}. Let 𝔟 = 𝔟_{1} ⊥ −𝔟_{2}. As 𝔟 is isotropic, 𝔟_{1} and 𝔟_{2} have a common nonzero value. Dividing the 𝔟_{i} by this nonzero common value, we may assume that the 𝔟_{i} are Pfister forms. We have

𝔟 ≃ 𝔠 ⊗ (𝔡′_{1} ⊥ −𝔡′_{2}) ⊥ (𝔠 ⊥ −𝔠).

The form 𝔠 ⊥ −𝔠 is metabolic by Example 1.22(2) and 𝔦(𝔟) > dim 𝔠. Therefore, the form 𝔠 ⊗ (𝔡′_{1} ⊥ −𝔡′_{2}) is isotropic, hence there is a ∈ D(𝔠 ⊗ 𝔡′_{1}) ∩ D(𝔠 ⊗ 𝔡′_{2}). By Proposition 6.15, we have 𝔟_{1} ≃ 𝔠 ⊗ ⟨⟨−a⟩⟩ ⊗ 𝔢_{1} and 𝔟_{2} ≃ 𝔠 ⊗ ⟨⟨−a⟩⟩ ⊗ 𝔢_{2} for some bilinear Pfister forms 𝔢_{1} and 𝔢_{2}. As

𝔟 ≃ 𝔠 ⊗ (𝔢′_{1} ⊥ −𝔢′_{2}) ⊥ (𝔠 ⊗ ⟨⟨−a⟩⟩ ⊥ −𝔠 ⊗ ⟨⟨−a⟩⟩),

either 𝔦(𝔟) = 2^{r+1} or we may repeat the argument. The result follows. □

If a general bilinear r-fold Pfister form 𝔠 is isometric to a common subform of two general Pfister forms 𝔟_{1} and 𝔟_{2}, we call it a linkage of 𝔟_{1} and 𝔟_{2} and say that 𝔟_{1} and 𝔟_{2} are r-linked. The integer m = max {r | 𝔟_{1} and 𝔟_{2} are r-linked} is called the linkage number of 𝔟_{1} and 𝔟_{2}. The proposition says that 𝔦(𝔟_{1} ⊥ −𝔟_{2}) = 2^{m}. If 𝔟_{1} and 𝔟_{2} are n-fold Pfister forms and r = n − 1, we say that 𝔟_{1} and 𝔟_{2} are linked. By Corollary 6.17 the linkage of any pair of bilinear Pfister forms is a divisor of each. The theory of linkage was first developed in [32].

If 𝔟 is a nondegenerate symmetric bilinear form over F, then the annihilator of 𝔟 in W (F) ,

ann_{W (F)}(𝔟) := {𝔠 ∈ W (F) | 𝔟 · 𝔠 = 0}

is an ideal in W (F). When 𝔟 is a Pfister form this ideal has a nice structure that we now establish. First note that if 𝔟 is an anisotropic Pfister form and x ∈ D(𝔟), then, as 𝔟 is round by Corollary 6.2, we have ⟨⟨x⟩⟩ ⊗ 𝔟 ≃ 𝔟 ⊥ −x𝔟 ≃ 𝔟 ⊥ −𝔟 is metabolic. It follows that ⟨⟨x⟩⟩ ∈ ann_{W (F)}(𝔟). We shall show that these binary forms generate ann_{W (F)}(𝔟). This will follow from the next result, also known as the Pfister-Witt Theorem.

Proposition 6.22. Let 𝔟 be an anisotropic bilinear Pfister form and 𝔠 a nondegenerate symmetric bilinear form. Then there exists a symmetric bilinear form 𝔡 satisfying all of the following:

(1) 𝔟 · 𝔠 = 𝔟 · 𝔡 in W (F).

(2) 𝔟 ⊗ 𝔡 is anisotropic. Moreover, dim 𝔡 ≤ dim 𝔠 and dim 𝔡 ≡ dim 𝔠 mod 2.

(3) 𝔠 − 𝔡 lies in the subgroup of W (F) generated by ⟨⟨x⟩⟩ with x ∈ D(𝔟).

Proof. We prove this by induction on dim 𝔠. By the Witt Decomposition Theorem 1.27, we may assume that 𝔠 is anisotropic. Hence 𝔠 is diagonalizable by Corollary 1.19, say 𝔠 = ⟨x_{1}, …, x_{n}⟩ with x_{i} ∈ F^{×}. If 𝔟 ⊗ 𝔠 is anisotropic, the result is trivial, so assume it is isotropic. Therefore, there exist a_{1}, …, a_{n} ∈ D̃(𝔟) not all zero such that a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0. Let b_{i} = a_{i} if a_{i} ≠ 0 and b_{i} = 1 otherwise. In particular, b_{i} ∈ G(𝔟) for all i. Let 𝔢 = ⟨b_{1}x_{1}, …, b_{n}x_{n}⟩. Then 𝔠 − 𝔢 = x_{1}⟨⟨b_{1}⟩⟩ + ⋯ + x_{n}⟨⟨b_{n}⟩⟩ with each b_{i} ∈ D(𝔟) as 𝔟 is round by Corollary 6.2. Since 𝔢 is isotropic, we have 𝔟 · 𝔠 = 𝔟 · (𝔢)_{an} in W (F). As dim(𝔢)_{an} < dim 𝔠, by the induction hypothesis there exists 𝔡 such that 𝔟 ⊗ 𝔡 is anisotropic and 𝔢 − 𝔡 and therefore 𝔠 − 𝔡 lies in the subgroup of W (F) generated by ⟨⟨x⟩⟩ with x ∈ D(𝔟). As 𝔟 ⊗ 𝔡 is anisotropic, it follows by (1) that dim 𝔡 ≤ dim 𝔠. It follows from (3) that the dimension of 𝔠 − 𝔡 is even. □

Corollary 6.23. Let 𝔟 be an anisotropic bilinear Pfister form. Then ann_{W (F)}(𝔟) is generated by ⟨⟨x⟩⟩ with x ∈ D(𝔟).

If 𝔟 is 2-dimensional, we obtain stronger results first established in [34].

Lemma 6.24. Let 𝔟 be a binary anisotropic bilinear form over F and 𝔠 an anisotropic bilinear form over F such that 𝔟 ⊗ 𝔠 is isotropic. Then 𝔠 ≃ 𝔡 ⊥ 𝔢 for some symmetric binary bilinear form 𝔡 annihilated by 𝔟 and symmetric bilinear form 𝔢 over F.

Proof. Let {e, f} be a basis for V_{𝔟}. By assumption there exist vectors v, w ∈ V_{𝔠} such that e ⊗ v + f ⊗ w is an isotropic vector for 𝔟 ⊗ 𝔠. Choose a 2-dimensional subspace W ⊂ V_{𝔠} containing v and w. Since 𝔠 is anisotropic, so is 𝔠|_{W}. In particular, 𝔠|_{W} is nondegenerate, hence 𝔠 = 𝔠|_{W} ⊥ 𝔠|_{W^{⊥}} by Proposition 1.6. As 𝔟 ⊗ (𝔠|_{W}) is an isotropic general 2-fold Pfister form, it is metabolic by Corollary 6.3. □

Proposition 6.25. Let 𝔟 be a binary anisotropic bilinear form over F and 𝔠 an anisotropic form over F. Then there exist symmetric bilinear forms 𝔠_{1} and 𝔠_{2} over F such that 𝔠 ≃ 𝔠_{1} ⊥ 𝔠_{2} with 𝔟 ⊗ 𝔠_{2} anisotropic and 𝔠_{1} ≃ 𝔡_{1} ⊥ ⋯ ⊥ 𝔡_{n} where each 𝔡_{i} is a binary bilinear form annihilated by 𝔟. In particular, if det 𝔡_{i} = −d_{i}F^{×2}, then d_{i} ∈ D(𝔟) for each i.

Proof. The first statement of the proposition follows from the lemma and the second from its proof. □

Corollary 6.26. Let 𝔟 be a binary anisotropic bilinear form over F and 𝔠 an anisotropic form over F annihilated by 𝔟. Then 𝔠 ≃ 𝔡_{1} ⊥ ⋯ ⊥ 𝔡_{n} for some symmetric binary forms 𝔡_{i} annihilated by 𝔟 for i ∈ [1, n].

CHAPTER II Quadratic Forms

7. Foundations

In this section, we introduce the basic properties of quadratic forms over an arbitrary field F. Their study arose from the investigation of homogeneous polynomials of degree two. If the characteristic of F is different from 2, then this study and that of symmetric bilinear forms are essentially the same as the diagonal of a symmetric bilinear form is a quadratic form and each determines the other by the polar identity. However, they are different when the characteristic of F is 2. In general, quadratic forms unlike bilinear forms have a rich geometric flavor. When studying symmetric bilinear forms, we saw that one could easily reduce to the study of nondegenerate forms. For quadratic forms, the situation is more complex. The polar form of a quadratic form no longer determines the quadratic form when the underlying field is of characteristic 2. However, the radical of the polar form is invariant under field extension. This leads to two types of quadratic forms. One is the study of totally singular quadratic forms, i.e., those whose polar bilinear form is zero. Such quadratic forms need not be trivial in the case of characteristic 2. The other extreme is when the radical of the polar form is as small as possible (which means of dimension zero or one), this gives rise to nondegenerate quadratic forms. As in the study of bilinear forms, certain properties are not invariant under base extension. The most important of these is anisotropy. Analogous to the bilinear case, an anisotropic quadratic form is one having no nontrivial zero, i.e., no isotropic vectors. Every vector that is isotropic for the quadratic form is isotropic for its polar form. If the characteristic is 2, the converse is false as every vector is an isotropic vector of the polar form. As in the previous chapter, we shall base this study on a coordinate free approach and strive to give uniform proofs in a characteristic free fashion.

Definition 7.1. Let V be a finite dimensional vector space over F. A quadratic form on V is a map φ : V → F satisfying:

(1) φ(av) = a^{2}φ(v) for all v ∈ V and a ∈ F.

(2) (Polar Identity) 𝔟_{φ} : V × V → F defined by

𝔟_{φ}(v, w) = φ(v + w) − φ(v) − φ(w)

is a bilinear form.

The bilinear form 𝔟_{φ} is called the polar form of of φ. We call dim V the dimension of the quadratic form and also write it as dim φ. We write φ is a quadratic form over F if φ is a quadratic form on a finite dimensional vector space over F and denote the underlying space by V_{φ}.

Note that the polar form of a quadratic form is automatically symmetric and even alternating if char F = 2. If 𝔟 : V × V → F is a bilinear form (not necessarily symmetric), let φ_{𝔟} : V → F be defined by φ_{𝔟}(v) = 𝔟(v, v) for all v ∈ V. Then φ_{𝔟} is a quadratic form and its polar form 𝔟_{φ_{𝔟}} is 𝔟 + 𝔟^{t}. We call φ_{𝔟} the associated quadratic form of 𝔟.

In particular, if 𝔟 is symmetric, the composition 𝔟 ↦ φ_{𝔟} ↦ 𝔟_{φ_{𝔟}} is multiplication by 2 as is the composition φ ↦ 𝔟_{φ} ↦ φ_{𝔟_{φ}}.

Let φ and ψ be two quadratic forms. An isometry f : φ → ψ is a linear map f : V_{φ} → V_{ψ} such that φ(v) = ψ(f(v)) for all v ∈ V_{φ}. If such an isometry exists, we write φ ≃ ψ and say that φ and ψ are isometric.

Example 7.2. If φ is a quadratic form over F and v ∈ V satisfies φ(v) ≠ 0, then the (hyperplane) reflection

τ_{v} : φ → φ given by w ↦ w − 𝔟_{φ}(v, w)φ(v)^{−1}v

is an isometry.

Let V be a finite dimensional vector space over F. Define the hyperbolic form of V to be the form ℍ(V) = φ_{ℍ} on V ⊕ V^{∗} defined by

φ_{ℍ}(v, f) := f(v)

for all v ∈ V and f ∈ V^{∗}. Note that the polar form of φ_{ℍ} is 𝔟_{φ_{ℍ}} = ℍ_{1}(V). If φ is a quadratic form isometric to ℍ(W) for some vector space W, we call φ a hyperbolic form. The form ℍ(F) is called the hyperbolic plane and we denote it simply by ℍ. If φ ≃ ℍ, two vectors e, f ∈ V_{φ} satisfying φ(e) = φ(f) = 0 and 𝔟_{φ}(e, f) = 1 are called a hyperbolic pair.

Let φ be a quadratic form on V and let {v_{1}, …, v_{n}} be a basis for V. Let a_{ii} = φ(v_{i}) for all i and

a_{ij} = \begin{cases}𝔟_{φ}(v_{i}, v_{j}) & for all i < j, \\ 0 & for all i > j. \end{cases}

As

φ(∑_{i=1}^{n} x_{i}v_{i}) = ∑_{i,j} a_{ij}x_{i}x_{j},

the homogeneous polynomial on the right hand side as well as the matrix (a_{ij}) determined by φ completely determines φ.

Notation 7.3. (1) Let a ∈ F. The quadratic form on F given by φ(v) = av^{2} for all v ∈ F will be denoted by ⟨a⟩_{q} or simply ⟨a⟩.

(2) Let a, b ∈ F. The 2-dimensional quadratic form on F^{2} given by φ(x, y) = ax^{2} + xy + by^{2} will be denoted by [a, b]. The corresponding matrix for φ in the standard basis is

A = (\begin{matrix}a & 1 \\ 0 & b \end{matrix}) ,

while the corresponding matrix for 𝔟_{φ} is

(\begin{matrix}2a & 1 \\ 1 & 2b \end{matrix}) = A + A^{t}.

Remark 7.4. Let φ be a quadratic form on V over F. Then the associated polar form 𝔟_{φ} is not the zero form if and only if there are two vectors v, w in V satisfying b(v, w) = 1. In particular, if φ is a nonzero binary form, then φ ≃ [a, b] for some a, b ∈ F.

Example 7.5. Let {e, f} be a hyperbolic pair for ℍ. Using the basis {e, ae + f}, we have ℍ ≃ [0, 0] ≃ [0, a] for any a ∈ F.

Example 7.6. Let char F = 2 and ℘ : F → F be the Artin-Schreier map ℘(x) = x^{2}+x. Let a ∈ F. Then the quadratic form [1, a] is isotropic if and only if a ∈ ℘(F).

Let V be a finite dimension vector space over F. The set Quad(V) of quadratic forms on V is a vector space over F. We have linear maps

Bil(V) → Quad(V) given by 𝔟 ↦ φ_{𝔟}

and

Quad(V) → Sym(V) given by φ ↦ 𝔟_{φ}.

Restricting the first map to Sym(V) and composing shows the compositions

Sym(V) → Quad(V) → Sym(V) and Quad(V) → Sym(V) → Quad(V)

are multiplication by 2. In particular, if char F ≠ 2 the map Quad(V) → Sym(V) given by φ ↦ \frac{1}{2}𝔟_{φ} is an isomorphism inverse to the map Sym(V) → Quad(V) by 𝔟 ↦ φ_{𝔟}. For this reason, we shall usually identify quadratic forms and symmetric bilinear forms over a field of characteristic different from 2.

The correspondence between quadratic forms on a vector space V of dimension n and matrices defines a linear isomorphism Quad(V) → 𝕋_{n}(F), where 𝕋_{n}(F) is the vector space of n×n upper-triangular matrices. Therefore by the surjectivity of the linear epimorphism 𝕄_{n}(F) → 𝕋_{n}(F) given by (a_{ij}) ↦ (b_{ij}) with b_{ij} = a_{ij} + a_{ji} for all i < j, and b_{ii} = a_{ii} for all i, and b_{ij} = 0 for all j < i implies that the linear map Bil(V) → Quad(V) given by 𝔟 ↦ φ_{𝔟} is also surjective. We, therefore, have an exact sequence

0 → Alt(V) → Bil(V) → Quad(V) → 0.

Exercise 7.7. The natural exact sequence

0 → Λ^{2}(V^{∗}) → V^{∗} ⊗_{F} V^{∗} → S^{2}(V^{∗}) → 0

can be identified with the sequence above via the isomorphism

S^{2}(V^{∗}) → Quad(V) given by f · g ↦ φ_{f·g} : v ↦ f(v)g(v).

If φ, ψ ∈ Quad(V), we say φ is similar to ψ if there exists an a ∈ F^{×} such that φ ≃ aψ.

Let φ be a quadratic form on V. A vector v ∈ V is called anisotropic if φ(v) ≠ 0 and isotropic if v ≠ 0 and φ(v) = 0. We call φ anisotropic if there are no isotropic vectors in V and isotropic if there are.

If W ⊂ V is a subspace, the restriction of φ on W is the quadratic form whose polar form is given by 𝔟_{φ|_{W}} = 𝔟_{φ}|_{W}. It is denoted by φ|_{W} and called a subform of φ. Define W^{⊥} to be the orthogonal complement of W relative to the polar form of φ. The space W is called a totally isotropic subspace if φ|_{W} = 0. If this is the case, then 𝔟_{φ}|_{W} = 0.

Example 7.8. If F is algebraically closed, then any homogeneous polynomial in more than one variable has a nontrivial zero. In particular, up to isometry, the only anisotropic quadratic forms over F are 0 and ⟨1⟩.

Remark 7.9. Let φ be a quadratic form on V over F. If φ = φ_{𝔟} for some symmetric bilinear form 𝔟, then φ is isotropic if and only if 𝔟 is. In addition, if char F ≠ 2, then φ is isotropic if and only if its polar form 𝔟_{φ} is. However, if char F = 2, then every 0 ≠ v ∈ V is an isotropic vector for 𝔟_{φ}.

Let ψ be a subform of a quadratic form φ. The restriction of φ on (V_{ψ})^{⊥} (with respect to the polar form 𝔟_{φ}) is denoted by ψ^{⊥} and is called the complementary form of ψ in φ. If V_{φ} = W ⊕ U is a direct sum of vector spaces with W ⊂ U^{⊥}, we write φ = φ|_{W} ⊥ φ|_{U} and call it an internal orthogonal sum. So φ(w + u) = φ(w) + φ(u) for all w ∈ W and u ∈ U. Note that φ|_{U} is a subform of (φ|_{W})^{⊥}.

Remark 7.10. Let φ be a quadratic form with rad 𝔟_{φ} = 0. If ψ is a subform of φ, then by Proposition 1.5, we have dim ψ^{⊥} = dim φ − dim ψ and therefore ψ^{⊥⊥} = ψ.

Let φ be a quadratic form on V. We say that φ is totally singular if its polar form 𝔟_{φ} is zero. If char F ≠ 2, then φ is totally singular if and only if φ is the zero quadratic form. If char F = 2 this may not be true. Define the quadratic radical of φ by

rad φ := {v ∈ rad 𝔟_{φ} | φ(v) = 0}.

This is a subspace of rad 𝔟_{φ}. We say that φ is regular if rad φ = 0. If char F ≠ 2, then rad φ = rad 𝔟_{φ}. In particular, φ is regular if and only if its polar form is nondegenerate. If char F = 2, this may not be true.

Example 7.11. Every anisotropic quadratic form is regular.

Clearly, if f : φ → ψ is an isometry of quadratic forms, then f(rad 𝔟_{φ}) = rad 𝔟_{ψ} and f(rad φ) = rad ψ.

Let φ be a quadratic form on V and : V → \overline{V} = V/ rad φ the canonical epimorphism. Let \overline{φ} denote the quadratic form on \overline{V} given by \overline{φ}(\overline{v}) := φ(v) for all v ∈ V. In particular, the restriction of \overline{φ} to rad 𝔟_{φ}/ rad φ determines an anisotropic quadratic form. We have:

Lemma 7.12. Let φ be a quadratic form on V and W any subspace of V satisfying V = rad φ ⊕ W. Then

φ = φ|_{rad φ} ⊥ φ|_{W} = 0|_{rad φ} ⊥ φ|_{W}

with φ|_{W} ≃ \overline{φ} the induced quadratic form on V/ rad φ. In particular, φ|_{W} is unique up to isometry.

If φ is a quadratic form, the form φ|_{W}, unique up to isometry will be called its regular part. The subform φ|_{W} in the lemma is regular but 𝔟_{φ|_{W}} may be degenerate if char F = 2. To obtain a further orthogonal decomposition of a quadratic form, we need to look at the regular part. The key is the following:

Proposition 7.13. Let φ be a regular quadratic form on V. Suppose that V contains an isotropic vector v. Then there exists a 2-dimensional subspace W of V containing v such that φ|_{W} ≃ ℍ.

Proof. As rad φ = 0, we have v ∉ rad 𝔟_{φ}. Thus there exists a vector w ∈ V such that a = 𝔟_{φ}(v, w) ≠ 0. Replacing v by a^{−1}v, we may assume that a = 1. Let W = F v ⊕ F w. Then v, w − φ(w)v is a hyperbolic pair. □

We say that any isotropic regular quadratic form splits off a hyperbolic plane. If K/F is a field extension, let φ_{K} be the quadratic form on V_{K} defined by

φ_{K}(x ⊗ v) := x^{2}φ(v) for all x ∈ K and v ∈ V with polar form 𝔟_{φ_{K}} = (𝔟_{φ})_{K}. Although (rad 𝔟_{φ})_{K} = rad(𝔟_{φ})_{K}, we only have (rad φ)_{K} ⊂ rad φ_{K} with inequality possible.

Remark 7.14. If K/F is a field extension and φ a quadratic form over F, then φ is regular if φ_{K} is.

The following is a useful observation. The proof analogous to that for Lemma 1.21 shows:

Lemma 7.15. Let φ be an anisotropic quadratic form over F. If K/F is purely transcendental, then φ_{K} is anisotropic.

7.A. Nondegenerate quadratic forms. To define nondegeneracy, we use the following lemma.

Lemma 7.16. Let φ be a quadratic form over F. Then the following are equivalent: (1) φ_{K} is regular for every field extension K/F.

(2) φ_{K} is regular over an algebraically closed field K containing F.

(3) φ is regular and dim rad 𝔟_{φ} ≤ 1.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3): As (rad φ)_{K} ⊂ rad φ_{K} = 0, we have rad φ = 0. To show the second statement, we may assume that F is algebraically closed. As φ|_{rad 𝔟_{φ}} = \overline{φ}|_{rad 𝔟_{φ}/ rad φ} is anisotropic and over an algebraically closed field, any quadratic form of dimension greater than one is isotropic, dim rad 𝔟_{φ} ≤ 1.

(3) ⇒ (1): Suppose that rad φ_{K} ≠ 0. As rad φ_{K} ⊂ rad 𝔟_{φ_{K}} and rad 𝔟_{φ_{K}} = (rad 𝔟_{φ})_{K} is of dimension at most one, we have rad φ_{K} = (rad 𝔟_{φ})_{K}. Let 0 ≠ v ∈ rad 𝔟_{φ}. Then v ∈ rad φ_{K}, hence φ(v) = 0 contradicting rad φ = 0. □

Definition 7.17. A quadratic form φ over F is called nondegenerate if the equivalent conditions of the lemma are satisfied.

Remark 7.18. If K/F is a field extension, then φ is nondegenerate if and only if φ_{K} is nondegenerate by Lemma 7.16.

This definition of a nondegenerate quadratic form agrees with the one given in [86]. It is different than that found in some other texts. The geometric characterization of this definition of nondegeneracy explains our definition. In fact, if φ is a nonzero quadratic form on V of dimension at least two, then the following are equivalent:

(1) The quadratic form φ is nondegenerate.

(2) The projective quadric X_{φ} associated to φ is smooth. (Cf. Proposition 22.1.)

(3) The even Clifford algebra C_{0}(φ) (cf. §11 below) of φ is separable (i.e., is a product of finite dimensional simple algebras each central over a separable field extension of F). (Cf. Proposition 11.6.)

(4) The group scheme SO(φ) of all isometries of φ identical on rad φ is reductive (semi-simple if dim φ ≥ 3 and simple if dim φ ≥ 5). (Cf. [86, Chapter VI].)

Proposition 7.19. (1) The form ⟨a⟩ is nondegenerate if and only if a ∈ F^{×}.

(2) The form [a, b] is nondegenerate if and only if 1 − 4ab ≠ 0. In particular, this binary quadratic form as well as its polar form is always nondegenerate if char F = 2.

(3) Hyperbolic forms are nondegenerate.

(4) Every binary isotropic nondegenerate quadratic form is isomorphic to ℍ.

Proof. (1) and (3) are clear.

(2): This follows by computing the determinant of the matrix representing the polar form corresponding to [a, b]. (Cf. Notation 7.3.)

(4) follows by Proposition 7.13. □

Let φ_{i} be a quadratic form on V_{i} for i = 1, 2. Then their external orthogonal sum, denoted by φ_{1} ⊥ φ_{2}, is the form on V_{1} ⊕ V_{2} given by

(φ_{1} ⊥ φ_{2})((v_{1}, v_{2})) := φ_{1}(v_{1}) + φ_{2}(v_{2})

for all v_{i} ∈ V_{i}, i = 1, 2. Note that 𝔟_{φ_{1}⊥φ_{2}} = 𝔟_{φ_{1}} ⊥ 𝔟_{φ_{2}}.

Remark 7.20. Let char F ≠ 2. Let φ and ψ be quadratic forms over F. (1) The form φ is nondegenerate if and only if φ is regular.

(2) If φ and ψ are both nondegenerate, then φ ⊥ ψ is nondegenerate as 𝔟_{φ⊥ψ} = 𝔟_{φ} ⊥ 𝔟_{ψ}.

Remark 7.21. Let char F = 2. Let φ and ψ be quadratic forms over F.

(1) If dim φ is even, then φ is nondegenerate if and only if its polar form 𝔟_{φ} is nondegenerate.

(2) If dim φ is odd, then φ is nondegenerate if and only if dim rad 𝔟_{φ} = 1 and φ|_{rad 𝔟_{φ}} is nonzero.

(3) If φ and ψ are nondegenerate quadratic forms over F at least one of which is of even dimension, then φ ⊥ ψ is nondegenerate.

The important analogue of Proposition 1.6 is immediate (using Lemma 7.16 for the last statement):

Proposition 7.22. Let φ be a quadratic form on V. Let W be a vector subspace such that 𝔟_{φ|_{W}} is a nondegenerate bilinear form. Then φ|_{W} is nondegenerate and φ = φ|_{W} ⊥ φ|_{W^{⊥}}. In particular, (φ|_{W})^{⊥} = φ|_{W^{⊥}}. Further, if φ is also nondegenerate, then so is φ|_{W^{⊥}}.

Example 7.23. Suppose that char F = 2 and a, b, c ∈ F. Let φ = [c, a] ⊥ [c, b] and {e, f, e′, f′} be a basis for V_{φ} satisfying φ(e) = c = φ(e′), φ(f) = a, φ(f′) = b, and 𝔟_{φ}(e, f) = 1 = 𝔟(e′, f′). Then in the basis {e, f + f′, e + e′, f′}, we have

[c, a] ⊥ [c, b] ≃ [c, a + b] ⊥ ℍ

by Example 7.5.

If n is a nonnegative integer and φ is a quadratic form over F, we let

nφ := \underbrace{φ ⊥ ⋯ ⊥ φ}_{n} .

In particular, if n is an integer, we do not interpret nφ with n viewed in the field. For example, if V is an n-dimensional vector space, ℍ(V) ≃ nℍ.

We denote ⟨a_{1}⟩_{q} ⊥ ⋯ ⊥ ⟨a_{n}⟩_{q} by

⟨a_{1}, …, a_{n}⟩_{q} or simply ⟨a_{1}, …, a_{n}⟩.

So φ ≃ ⟨a_{1}, …, a_{n}⟩ for some a_{i} ∈ F if and only if V_{φ} has an orthogonal basis. If V_{φ} has an orthogonal basis, we say φ is diagonalizable.

Remark 7.24. Suppose that char F = 2 and φ is a quadratic form over F. Then φ is diagonalizable if and only if φ is totally singular, i.e., its polar form 𝔟_{φ} = 0. If this is the case, then every basis for V_{φ} is orthogonal. In particular, there are no diagonalizable nondegenerate quadratic forms of dimension greater than one.

Exercise 7.25. A quadratic form φ is diagonalizable if and only if φ = φ_{𝔟} for some symmetric bilinear form 𝔟.

Example 7.26. Suppose that char F ≠ 2. If a ∈ F^{×}, then ⟨a, −a⟩ ≃ ℍ.

Example 7.27. (Cf. Example 1.10.) Let char F = 2 and φ = ⟨1, a⟩ with a ≠ 0. If {e, f} is the basis for V_{φ} with φ(e) = 1 and φ(f) = a, then computing on the orthogonal basis {e, xe + yf} with x, y ∈ F, y ≠ 0 shows φ ≃ ⟨1, x^{2} + ay^{2}⟩. Consequently, ⟨1, a⟩ ≃ ⟨1, b⟩ if and only if b = x^{2} + ay^{2} with y ≠ 0.

7.B. Structure theorems for quadratic forms. We wish to decompose a quadratic form over a field F into an orthogonal sum of nice subforms. We begin with nondegenerate quadratic forms with large totally isotopic subspaces. Unlike the case of symmetric bilinear forms in characteristic 2, 2-dimensional nondegenerate isotropic quadratic forms are hyperbolic.

Proposition 7.28. Let φ be an 2n-dimensional nondegenerate quadratic form on V. Suppose that V contains a totally isotropic subspace W of dimension n. Then φ ≃ nℍ. Conversely, every hyperbolic form of dimension 2n contains a totally isotropic subspace of dimension n.

Proof. Let 0 ≠ v ∈ W. Then by Proposition 7.13 there exists a 2-dimensional subspace V_{1} of V containing v with φ|_{V_{1}} a nondegenerate subform isomorphic to ℍ. By Proposition 7.22, this subform splits off as an orthogonal summand. Since φ|_{V_{1}} is nondegenerate, W ∩ V_{1} is 1-dimensional, so dim W ∩ V_{1}^{⊥} = n − 1. The first statement follows by induction applied to the totally isotropic subspace W ∩ V_{1}^{⊥} of V_{1}^{⊥}. The converse is easy. □

We turn to splitting off anisotopic subforms of regular quadratic forms. It is convenient to write these decompositions separately for fields of characteristic 2 and not 2.

Proposition 7.29. Let char F ≠ 2 and φ a quadratic form on V over F. Then there exists an orthogonal basis for V. In particular, there exist 1-dimensional subspaces V_{i} ⊂ V, 1 ≤ i ≤ n for some n and an orthogonal decomposition

φ = φ|_{rad 𝔟_{φ}} ⊥ φ|_{V_{1}} ⊥ ⋯ ⊥ φ|_{V_{n}}

with φ|_{V_{1}} ≃ ⟨a_{i}⟩, a_{i} ∈ F^{×} for all i ∈ [1, n]. In particular,

φ ≃ r⟨0⟩ ⊥ ⟨a_{1}, …, a_{n}⟩

with r = dim rad 𝔟_{φ}.

Proof. We may assume that φ ≠ 0. Hence there exists an anisotropic vector 0 ≠ v ∈ V. As 𝔟_{φ|_{F v}} is nondegenerate, φ|_{F v} splits off as an orthogonal summand of φ by Proposition 7.22. The result follows easily by induction. □

Corollary 7.30. Suppose that char F ≠ 2. Then every quadratic form over F is diagonalizable.

Proposition 7.31. Let char F = 2 and φ a quadratic form on V over F. Then there exists 2-dimensional subspaces V_{i} ⊂ V, 1 ≤ i ≤ n for some n, a subspace W ⊂ rad 𝔟_{φ}, and an orthogonal decomposition

φ = φ|_{rad(φ)} ⊥ φ|_{W} ⊥ φ|_{V_{1}} ⊥ ⋯ ⊥ φ|_{V_{n}}

with φ|_{V_{i}} ≃ [a_{i}, b_{i}] nondegenerate, a_{i}, b_{i} ∈ F for all i ∈ [1, n]. Moreover, φ|_{W} is anisotropic, diagonalizable, and is unique up to isometry. In particular,

φ ≃ r⟨0⟩ ⊥ ⟨c_{1}, …, c_{s}⟩ ⊥ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{n}, b_{n}]

with r = dim rad φ and s = dim W, c_{i} ∈ F^{×}, 1 ≤ i ≤ s.

Proof. Let W ⊂ V be a subspace such that rad 𝔟_{φ} = rad φ ⊕ W and V′ ⊂ V a subspace such that V = rad 𝔟_{φ} ⊕ V′. Then φ = φ|_{rad(φ)} ⊥ φ|_{W} ⊥ φ|_{V′}. The form φ|_{W} is diagonalizable as 𝔟_{φ|_{W}} = 0 and anisotropic as W ∩ rad φ = 0. By Lemma 7.12, the form φ|_{W} = (φ|_{rad 𝔟_{φ}})|_{W} is unique up to isometry. So to finish we need only show that φ|_{V′} is an orthogonal sum of nondegenerate binary subforms of the desired isometry type. We may assume that V′ ≠ {0}. Let 0 ≠ v ∈ V′. Then there exists 0 ≠ v′ ∈ V′ such that c = 𝔟_{φ}(v, v′) ≠ 0. Replacing v′ by c^{−1}v′, we may assume that 𝔟_{φ}(v, v′) = 1. In particular, φ|_{F v⊕F v′} ≃ [φ(v), φ(v′)]. As [φ(v), φ(v′)] and its polar form are nondegenerate by Proposition 7.19, the subform φ|_{F v⊕F v′} is an orthogonal direct summand of φ by Proposition 7.22. The decomposition follows by Lemma 7.12 and induction. □

Corollary 7.32. Let char F = 2 and let φ be a nondegenerate quadratic form over F.

(1) If dim φ = 2n, then

φ ≃ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{n}, b_{n}]

for some a_{i}, b_{i} ∈ F, i ∈ [1, n].

(2) If dim φ = 2n + 1, then

φ ≃ ⟨c⟩ ⊥ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{n}, b_{n}]

for some a_{i}, b_{i} ∈ F, i ∈ [1, n], and c ∈ F^{×} unique up to F^{×2}.

Example 7.33. Suppose that F is quadratically closed of characteristic 2. Then every anisotropic form is isometric to 0, ⟨1⟩ or [1, a] with a ∈ F \ ℘(F) where ℘ : F → F is the Artin-Schreier map.

Exercise 7.34. Every nondegenerate quadratic form over a separably closed field F is isometric to nℍ or ⟨a⟩ ⊥ nℍ for some n ≥ 0 and a ∈ F^{×}.

8. Witt’s Theorems

As with the bilinear case, the classical Witt theorems are more delicate to ascertain over fields of arbitrary characteristic. We shall give characteristic free proofs of these. The basic Witt theorem is the Witt Extension Theorem (cf. Theorem 8.3 below). Witt’s original theorem (cf. [139]) has been generalized in various ways. We use one similar to that given by Kneser (cf. [80, Th. 1.2.2]). We construct the quadratic Witt group of even-dimensional anisotropic quadratic forms and use the Witt theorems to study this group.

To obtain further decompositions of a quadratic form, we need generalizations of the classical Witt theorems for bilinear forms over fields of characteristic different from 2.

Let φ be a quadratic form on V. Let v and v′ in V satisfy φ(v) = φ(v′). If the vector v̄ = v − v′ is anisotropic, then the reflection (cf. Example 7.2) τv̄ : φ → φ satisfies

(8.1)  τv̄(v) = v′.

What if v̄ is isotropic?

Lemma 8.2. Let φ be a quadratic form on V with polar form 𝔟. Let v and v′ lie in V and v̄ = v − v′. Suppose that φ(v) = φ(v′) and φ(v̄) = 0. If w ∈ V is anisotropic and satisfies that both 𝔟(w, v) and 𝔟(w, v′) are nonzero, then the vector w′ = v − τ_{w}(v′) is anisotropic and (τ_{w} ∘ τ_{w′})(v) = v′.

Proof. As w′ = v̄ + 𝔟(v′, w)φ(w)^{−1}w, we have
φ(w′) = φ(v̄) + 𝔟(v̄, 𝔟(v′, w)φ(w)^{−1}w) + 𝔟(v′, w)^{2}φ(w)^{−1}
= 𝔟(v, w)𝔟(v′, w)φ(w)^{−1} ≠ 0.

It follows from (8.1) that τ_{w′}(v) = τ_{w}(v′), hence the result. □

Theorem 8.3 (Witt Extension Theorem). Let φ and φ′ be isometric quadratic forms on V and V′ respectively. Let W ⊂ V and W′ ⊂ V′ be subspaces such that W ∩ rad 𝔟_{φ} = 0 and W′ ∩ rad 𝔟_{φ′} = 0. Suppose that there is an isometry α : φ|_{W} → φ′|_{W′}. Then there exists an isometry α̃ : φ → φ′ such that α̃(W) = W′ and α̃|_{W} = α.

Proof. It is sufficient to treat the case V = V′ and φ = φ′. Let 𝔟 denote the polar form of φ. We proceed by induction on n = dim W, the case n = 0 being obvious. Suppose that n > 0. In particular, φ is not identically zero. Let u ∈ V satisfy φ(u) ≠ 0. As dim W ∩ (F u)^{⊥} ≥ n − 1, there exists a subspace W_{0} ⊂ W of codimension one with W_{0} ⊂ (F u)^{⊥}. Applying the induction hypothesis to β = α|_{W_{0}} : φ|_{W_{0}} → φ|_{α(W_{0})}, there exists an isometry β̃ : φ → φ satisfying β̃(W_{0}) = α(W_{0}) and β̃|_{W_{0}} = β. Replacing W′ by β̃^{−1}(W′), we may assume that W_{0} ⊂ W′ and α|_{W_{0}} is the identity.

Let v be any vector in W \ W_{0} and set v′ = α(v) ∈ W′. It suffices to find an isometry γ of φ such that γ(v) = v′ and γ|_{W_{0}} = Id, the identity on W_{0}. Let v̄ = v − v′ as above and S = W_{0}^{⊥}. Note that for every w ∈ W_{0}, we have α(w) = w, hence

𝔟(v̄, w) = 𝔟(v, w) − 𝔟(α(v), α(w)) = 0,

i.e., v̄ ∈ S.

Suppose that φ(v̄) ≠ 0. Then τv̄(v) = v′ using (8.1). Moreover, τv̄(w) = w for every w ∈ W_{0} as v̄ is orthogonal to W_{0}. Then γ = τv̄ works. So we may assume that φ(v̄) = 0. We have

0 = φ(v̄) = φ(v) − 𝔟(v, v′) + φ(v′) = 𝔟(v, v) − 𝔟(v, v′) = 𝔟(v, v̄),

i.e., v̄ is orthogonal to v. Similarly, v̄ is orthogonal to v′.

By Proposition 1.5, the map l_{W} : V → W^{∗} is surjective. In particular, there exists u ∈ V such that 𝔟(u, W_{0}) = 0 and 𝔟(u, v) = 1. In other words, v is not orthogonal to S, i.e., the intersection H = (F v)^{⊥} ∩ S is a subspace of codimension one in S. Similarly, H′ = (F v′)^{⊥} ∩ S is also a subspace of codimension one in S. Note that v̄ ∈ H ∩ H′.

Suppose that there exists an anisotropic vector w ∈ S such that w ∉ H and w ∉ H′. By Lemma 8.2, we have (τ_{w} ∘ τ_{w′})(v) = v′ where

w′ = v − τ_{w}(v′) = v̄ + 𝔟(v′, w)φ(w)^{−1}w ∈ S.

As w, w′ ∈ S, the map τ_{w} ∘ τ_{w′} is the identity on W_{0}. Setting γ = τ_{w} ∘ τ_{w′} produces the desired extension. Consequently, we may assume that φ(w) = 0 for every w ∈ S \ (H ∪ H′).

Case 1: |F | > 2.

Let w_{1} ∈ H ∩ H′ and w_{2} ∈ S \ (H ∪ H′). Then aw_{1} + w_{2} ∈ S \ (H ∪ H′) for any a ∈ F, so by assumption

0 = φ(aw_{1} + w_{2}) = a^{2}φ(w_{1}) + a𝔟(w_{1}, w_{2}) + φ(w_{2}).

Since |F | > 2, we must have φ(w_{1}) = 𝔟(w_{1}, w_{2}) = φ(w_{2}) = 0. So φ(H ∩ H′) = 0, φ(S \ (H ∪ H′)) = 0, and H ∩ H′ is orthogonal to S \ (H ∪ H′), (i.e., 𝔟(x, y) = 0 for all x ∈ H ∩ H′ and y ∈ S \ (H ∪ H′)).

Let w ∈ H and w′ ∈ S\(H ∪H′). As |F | > 2, we see that w+aw′ ∈ S\(H ∪H′) for some a ∈ F. Hence the set S \ (H ∪ H′) generates S. Consequently, H ∩ H′ is orthogonal to S. In particular, 𝔟(v̄, S) = 0. Thus H = H′. It follows that φ(H) = 0 and φ(S \ H) = 0, hence φ(S) = 0, a contradiction. This finishes the proof in this case.

Case 2: F = 𝔽_{2}, the field of two elements.

As H ∪ H′ ≠ S, there exists a w ∈ S such that 𝔟(w, v) ≠ 0 and 𝔟(w, v′) ≠ 0. As F = 𝔽_{2}, this means that 𝔟(w, v) = 1 = 𝔟(w, v′). Moreover, by our assumptions, φ(v̄) = 0 and φ(w) = 0. Consider the linear map

γ : V → V given by γ(x) = x + 𝔟(v̄, x)w + 𝔟(w, x)v̄.

Note that 𝔟(w, v̄) = 𝔟(w, v) + 𝔟(w, v′) = 1 + 1 = 0. A simple calculation shows that γ^{2} = Id and φ(γ(x)) = φ(x) for any x ∈ V, i.e., γ is an isometry. Moreover, γ(v) = v + v̄ = v′. Finally, γ|_{W_{0}} = Id since w and v̄ are orthogonal to W_{0}. □

Theorem 8.4 (Witt Cancellation Theorem). Let φ, φ′ be quadratic forms on V and V′ respectively, and ψ, ψ′ quadratic forms on W and W′ respectively, with rad 𝔟_{ψ} = 0 = rad 𝔟_{ψ′}. If

φ ⊥ ψ ≃ φ′ ⊥ ψ′ and ψ ≃ ψ′,

then φ ≃ φ′.

Proof. Let f : ψ → ψ′ be an isometry. By the Witt Extension Theorem, f extends to an isometry f̃: φ ⊥ ψ → φ′ ⊥ ψ′. As f̃ takes V = W^{⊥} to V′ = (W′)^{⊥}, the result follows. □

Witt Cancellation together with our previous computations allow us to derive the decomposition that we want.

Theorem 8.5 (Witt Decomposition Theorem). Let φ be a quadratic form on V. Then there exist subspaces V_{1} and V_{2} of V such that φ = φ|_{rad φ} ⊥ φ|_{V_{1}} ⊥ φ|_{V_{2}} with φ|_{V_{1}} anisotropic and φ|_{V_{2}} hyperbolic. Moreover, φ|_{V_{1}} and φ|_{V_{2}} are unique up to isometry.

Proof. We know that φ = φ|_{rad φ} ⊥ φ|_{V′} with φ_{V′} on V′ unique up to isometry. Therefore, we can assume that φ is regular. Suppose that φ_{V′} is isotropic. By Proposition 7.13, we can split off a subform as an orthogonal summand isometric to the hyperbolic plane. The desired decomposition follows by induction. As every hyperbolic form is nondegenerate, the Witt Cancellation Theorem shows the uniqueness of φ|_{V_{1}} up to isometry hence φ|_{V_{2}} is unique by dimension count. □

8.A. Witt equivalence. Using the Witt Decomposition Theorem 8.5, we can define an equivalence of quadratic forms over a field F.

Definition 8.6. Let φ be a quadratic form on V and φ = φ|_{rad φ} ⊥ φ|_{V_{1}} ⊥ φ|_{V_{2}} be the decomposition in the theorem. The anisotropic form φ|_{V_{1}}, unique up to isometry, will be denoted φ_{an} and called the anisotropic part of φ. As φ_{V_{2}} is hyperbolic, dim V_{2} = 2n for some unique nonnegative number n. The integer n is called the Witt index of φ and denoted by 𝔦_{0}(φ). We say that two quadratic forms φ and ψ are Witt-equivalent and write φ ∼ ψ if dim rad φ = dim rad ψ and φ_{an} ≃ ψ_{an}. Equivalently, φ ∼ ψ if and only if φ ⊥ nℍ ≃ ψ ⊥ mℍ for some n and m.

Note that if φ ∼ ψ, then φ_{K} ∼ ψ_{K} for any field extension K/F.

Witt cancellation does not hold in general for nondegenerate quadratic forms in characteristic 2. We show in the next result, Proposition 8.8, that

(8.7)  [a, b] ⊥ ⟨a⟩ ≃ ℍ ⊥ ⟨a⟩

if char F = 2 for all a, b ∈ F with a ≠ 0. But [a, b] ≃ ℍ if and only if [a, b] is isotropic by Proposition 7.19(4). Although Witt cancellation does not hold in general in characteristic 2, we can use the following:

Proposition 8.8. Let ρ be a nondegenerate quadratic form of even dimension over a field F of characteristic 2. Then ρ ⊥ ⟨a⟩ ∼ ⟨a⟩ for some a ∈ F^{×} if and only if ρ ∼ [a, b] for some b ∈ F.

Proof. The case char F ≠ 2 is easy, so we can assume that char F = 2. Let φ = [a, b] ⊥ ⟨a⟩ with a, b ∈ F and a ≠ 0. Clearly, φ is isotropic and it is nondegenerate as φ|_{rad 𝔟_{φ}} = ⟨a⟩. It follows by Proposition 7.13 that [a, b] ⊥ ⟨a⟩ ≃ ℍ ⊥ ⟨a⟩ ∼ ⟨a⟩. Since ρ ∼ [a, b], we have ρ ⊥ ⟨a⟩ ∼ ⟨a⟩.

Conversely, suppose that ρ ⊥ ⟨a⟩ ∼ ⟨a⟩ for some a ∈ F^{×}. We prove the statement by induction on n = dim ρ. If n = 0, we can take b = 0. So assume that n > 0. We may also assume that ρ is anisotropic. By assumption, the form ρ ⊥ ⟨a⟩ is isotropic. Therefore, a = ρ(v) for some v ∈ V_{ρ}, and we can find a decomposition ρ = ρ′ ⊥ [a, d] for some nondegenerate form ρ′ of dimension n − 2 and b ∈ F. As [a, d] ⊥ ⟨a⟩ ≃ ℍ ⊥ ⟨a⟩ by the first part of the proof, we have

⟨a⟩ ∼ ρ ⊥ ⟨a⟩ = ρ′ ⊥ [a, d] ⊥ ⟨a⟩ ∼ ρ′ ⊥ ⟨a⟩.

By the induction hypothesis, ρ′ ≃ [a, c] for some c ∈ F. Therefore, by Example 7.23,

ρ = ρ′ ⊥ [a, d] ∼ [a, c] ⊥ [a, d] ≃ [a, c + d] ⊥ ℍ ∼ [a, c + d].  □

Remark 8.9. Let φ and ψ be quadratic forms over F.

(1) If φ is nondegenerate and anisotropic over F and K/F a purely transcendental extension, then φ_{K} remains anisotropic by Lemma 7.15. In particular, for any nondegenerate φ, we have 𝔦_{0}(φ) = 𝔦_{0}(φ_{K}).

(2) Let a ∈ F^{×}. Then φ ≃ aψ if and only if φ_{an} ≃ aψ_{an} as any form similar to a hyperbolic form is hyperbolic.

(3) If char F = 2, the quadratic form φ_{an} may be degenerate. This is not possible if char F ≠ 2.

(4) If char F ≠ 2, then every symmetric bilinear form corresponds to a quadratic form, hence the Witt theorems hold for symmetric bilinear forms in characteristic different from 2.

8.B. Totally isotropic subspaces. Given a regular quadratic form φ on V, we show that every totally isotopic subspace of V lies in a maximal isotropic subspace of V of dimension equal to the Witt index of φ.

Lemma 8.10. Let φ be a regular quadratic form on V with W ⊂ V a totally isotropic subspace of dimension m. If ψ is the quadratic form on W^{⊥}/W induced by the restriction of φ on W^{⊥}, then φ ≃ ψ ⊥ mℍ.

Proof. As W ∩ rad 𝔟_{φ} ⊂ rad φ, the intersection W ∩ rad 𝔟_{φ} is trivial. Thus the map V → W^{∗} by v ↦ l_{v}|_{W} : w ↦ 𝔟_{φ}(v, w) is surjective by Proposition 1.5 and dim W^{⊥} = dim V − dim W. Let W′ ⊂ V be a subspace mapping isomorphically onto W^{∗}. Clearly, W ∩ W′ = {0}. Let U = W ⊕ W′.

We show the form φ|_{U} is hyperbolic. The subspace W ⊕ W′ is nondegenerate with respect to 𝔟_{φ}. Indeed, let 0 ≠ v = w + w′ ∈ W ⊕ W′. If w′ ≠ 0, there exists a w_{0} ∈ W such that 𝔟_{φ}(w′, w_{0}) ≠ 0, hence 𝔟_{φ}(v, w_{0}) ≠ 0. If w′ = 0, there exists w′_{0} ∈ W′ such that 𝔟_{φ}(w, w′_{0}) ≠ 0, hence 𝔟_{φ}(v, w′_{0}) ≠ 0. Thus by Proposition 7.28, the form φ|_{U} is isometric to mℍ where m = dim W.

By Proposition 7.22, we have φ = φ|_{U^{⊥}} ⊥ φ|_{U} ≃ φ|_{U^{⊥}} ⊥ mℍ. As W and U^{⊥} are subspaces of W^{⊥} and U ∩ W^{⊥} = W, we have W^{⊥} = W ⊕ U^{⊥}. Thus W^{⊥}/W ≃ U^{⊥} and the result follows. □

Proposition 8.11. Let φ be a regular quadratic form on V. Then every totally isotropic subspace of V is contained in a totally isotropic subspace of dimension 𝔦_{0}(φ).

Proof. Let W ⊂ V be a totally isotropic subspace of V. We may assume that it is a maximal totally isotropic subspace. In the notation in the proof of Lemma 8.10, we have φ = φ|_{U^{⊥}} ⊥ φ|_{U} with φ|_{U} ≃ mℍ where m = dim W. The form φ|_{U^{⊥}} is anisotropic by the maximality of W, hence must be φ_{an} by the Witt Decomposition Theorem 8.5. In particular, dim W = 𝔦_{0}(φ). □

Corollary 8.12. Let φ be a regular quadratic form on V. Then every totally isotropic subspace W of V has dimension at most 𝔦_{0}(φ) with equality if and only if W is a maximal totally isotropic subspace of V.

Let ρ be a nondegenerate quadratic form and φ a subform of ρ. If 𝔟_{φ} is nondegenerate, then ρ = φ ⊥ φ^{⊥}, hence ρ ⊥ (−φ) ∼ φ^{⊥}. However, in general, ρ ≠ φ ⊥ φ^{⊥}. We do always have:

Lemma 8.13. Let ρ be a nondegenerate quadratic form of even dimension and φ a regular subform of ρ. Then ρ ⊥ (−φ) ∼ φ^{⊥}.

Proof. Let W be the subspace defined by W = {(v, v) | v ∈ V_{φ}} of V_{ρ} ⊕ V_{φ}. Clearly W is totally isotropic with respect to the form ρ ⊥ (−φ) on V_{ρ} ⊕ V_{φ}. By the proof of Lemma 8.10, we have dim W^{⊥}/W = dim V_{ρ} ⊕ V_{φ} − 2 dim W = dim V_{ρ} − dim V_{φ}. By Remark 7.10, we also have dim V_{φ}^{⊥} = dim V_{ρ} − dim V_{φ}. It follows that the linear map W^{⊥}/W → V_{φ}^{⊥} defined by (v, v′) ↦ v−v′ is an isometry. On the other hand, by Lemma 8.10, the form on W^{⊥}/W is Witt-equivalent to ρ ⊥ (−φ). □

Let V and W be vector spaces over F. Let 𝔟 be a symmetric bilinear form on W and φ be a quadratic form on V. The tensor product of 𝔟 and φ is the quadratic form 𝔟 ⊗ φ on W ⊗_{F} V defined by

(8.14)  (𝔟 ⊗ φ)(w ⊗ v) = 𝔟(w, w) · φ(v)

for all w ∈ W and v ∈ V with the polar form of 𝔟 ⊗ φ equal to 𝔟 ⊗ 𝔟_{φ}. For example, if a ∈ F, then ⟨a⟩_{b} ⊗ φ ≃ aφ.

Example 8.15. If 𝔟 is a symmetric bilinear form, then φ_{𝔟} ≃ 𝔟 ⊗ ⟨1⟩_{q}.

Lemma 8.16. Let 𝔟 be a nondegenerate symmetric bilinear form over F and φ a nondegenerate quadratic form over F. In addition, assume that dim φ is even if characteristic of F is 2. Then:

(1) The quadratic form 𝔟 ⊗ φ is nondegenerate.

(2) If either φ or 𝔟 is hyperbolic, then 𝔟 ⊗ φ is hyperbolic.

Proof. (1): The bilinear form 𝔟_{φ} is nondegenerate by Remark 7.20 and by Remark 7.21 if characteristic of F is not 2 or 2, respectively. By Lemma 2.1, the form 𝔟 ⊗ 𝔟_{φ} is nondegenerate, hence so is 𝔟 ⊗ φ.

(2): Using Proposition 7.28, we see that V_{𝔟⊗φ} contains a totally isotropic space of dimension \frac{1}{2} dim(𝔟 ⊗ φ). □

As the orthogonal sum of even-dimensional nondegenerate quadratic forms over F is nondegenerate, the isometry classes of even-dimensional nondegenerate quadratic forms over F form a monoid under orthogonal sum. The quotient of the Grothendieck group of this monoid by the subgroup generated by the image of the hyperbolic plane is called the quadratic Witt group and will be denoted by I_{q}(F). The tensor product of a bilinear with a quadratic form induces a W (F)-module structure on I_{q}(F) by Lemma 8.16.

Remark 8.17. Let φ and ψ be two nondegenerate even-dimensional quadratic forms over F. By the Witt Decomposition Theorem 8.5,

φ ≃ ψ if and only if φ = ψ in I_{q}(F) and dim φ = dim ψ.

Remark 8.18. Let F → K be a homomorphism of fields. Analogous to Proposition 2.7, this map induces the restriction homomorphism

r_{K/F} : I_{q}(F) → I_{q}(K).

It is a group homomorphism. If K/F is purely transcendental, the restriction map is injective by Lemma 7.15.

Suppose that char F ≠ 2. Then we have an isomorphism I(F) → I_{q}(F) given by 𝔟 ↦ φ_{𝔟}. We will use the correspondence 𝔟 ↦ φ_{𝔟} to identify bilinear forms in W (F) with quadratic forms. In particular, we shall view the class of a quadratic form in the Witt ring of bilinear forms when char F ≠ 2.

9. Quadratic Pfister forms I

As in the bilinear case, there is a special class of forms built from tensor products of forms. If the characteristic of F is different from 2, these forms can be identified with the bilinear Pfister forms. If the characteristic is 2, these forms arise as the tensor product of a bilinear Pfister form and a binary quadratic form of the type [1, a] and were first introduced by Baeza (cf. [15]). In general, the quadratic 1-fold Pfister forms are just the norm forms of a quadratic étale F -algebra and the 2-fold quadratic Pfister forms are just the reduced norm forms of quaternion algebras. These forms as their bilinear analogue satisfy the property of being round. In this section, we begin their study.

9.A. Values and similarities of quadratic forms. Analogously to the symmetric bilinear case, we study the values that a quadratic form can take as well as the similarity factors. We begin with some notation.

Let φ be a quadratic form on V over F. Let

D(φ) := {φ(v) | v ∈ V, φ(v) ≠ 0},

the set on nonzero values of φ and

G(φ) := {a ∈ F^{×} | aφ ≃ φ},

a group called the group of similarity factors of 𝔟. If D(φ) = F^{×}, we say that φ is universal. Also set

D̃(φ) := D(φ) ∪ {0}.

We say that elements in D̃(φ) are represented by φ. For example, G(ℍ) = F^{×} (as for bilinear hyperbolic planes) and D(ℍ) = F^{×}. In particular, if φ is an regular isotropic quadratic form over F, then φ is universal by Proposition 7.13.

The analogous proof of Lemma 1.13 shows:

Lemma 9.1. Let φ be a quadratic form. Then

D(φ) · G(φ) ⊂ D(φ).

In particular, if 1 ∈ D(φ), then G(φ) ⊂ D(φ).

The relationship between values and similarities of a symmetric bilinear form and its associated quadratic form is given by the following:

Lemma 9.2. Let 𝔟 a symmetric bilinear form on F and φ = φ_{𝔟}. Then: (1) D(φ) = D(𝔟).

(2) G(𝔟) ⊂ G(φ).

Proof. (1): By definition, φ(v) = 𝔟(v, v) for all v ∈ V.

(2): Let a ∈ G(𝔟) and λ : 𝔟 → a𝔟 an isometry. Then φ(λ(v)) = 𝔟(λ(v), λ(v)) = a𝔟(v, v) = aφ(v) for all v ∈ V. □

A quadratic form is called round if G(φ) = D(φ). In particular, if φ is round, then D(φ) is a group. For example, any hyperbolic form is round.

A basic example of round forms arises from quadratic F -algebras (cf. §98.B):

Example 9.3. Let K be a quadratic F -algebra. Then there exists an involution on K denoted by x ↦ x̄ and a quadratic norm form φ = N given by x ↦ xx̄ (cf. §98.B). We have φ(xy) = φ(x)φ(y) for all x, y ∈ K. If x ∈ K with φ(x) ≠ 0, then x ∈ K^{×}. Hence the map K → K given by multiplication by x is an F -isomorphism and φ(x) ∈ G(φ). Thus D(φ) ⊂ G(φ). As 1 ∈ D(φ), we have G(φ) ⊂ D(φ). In particular, φ is round.

Let K be a quadratic étale F -algebra. So K = F_{a} for some a ∈ F (cf. Examples 98.2 and 98.3). Denote the norm form N of F_{a} in Example 9.3 by ⟨⟨a]] and call it a quadratic 1-fold Pfister form. In particular, it is round. Explicitly, we have:

Example 9.4. For F_{a} a quadratic étale F algebra, we have:

(1) (Cf. Example 98.2.) If char F ≠ 2, then F_{a} = F [t]/(t^{2} − a) with a ∈ F^{×} and the quadratic form ⟨⟨a]] = ⟨1, −a⟩_{q} ≃ ⟨⟨a⟩⟩_{b} ⊗ ⟨1⟩_{q} is the norm form of F_{a}.

(2) (Cf. Example 98.3.) If char F = 2, then F_{a} = F [t]/(t^{2} + t + a) with a ∈ F and the quadratic form ⟨⟨a]] = [1, a] is the norm form of F_{a}. In particular, ⟨⟨a]] ≃ ⟨⟨x^{2} + x + a]] for any x ∈ F.

9.B. Quadratic Pfister forms and round forms. Let n ≥ 1. A quadratic form isometric to a quadratic form of the type

⟨⟨a_{1}, …, a_{n}]] := ⟨⟨a_{1}, …, a_{n−1}⟩⟩_{b} ⊗ ⟨⟨a_{n}]]

for some a_{1}, …, a_{n−1} ∈ F^{×} and a_{n} ∈ F (with a_{n} ≠ 0 if char F ≠ 2) is called a quadratic n-fold Pfister form. It is convenient to call the form isometric to ⟨1⟩_{q} a 0-fold Pfister form. Every quadratic n-fold Pfister form is nondegenerate by Lemma 8.16. We let

P_{n}(F) := {φ | φ a quadratic n-fold Pfister form},
P (F) := ⋃ P_{n}(F),
GP_{n}(F) := {aφ | a ∈ F^{×}, φ a quadratic n-fold Pfister form},
GP (F) := ⋃ GP_{n}(F).

Forms in GP_{n}(F) are called general quadratic n-fold Pfister forms.

If char F ≠ 2, the form ⟨⟨a_{1}, …, a_{n}]] is the associated quadratic form of the bilinear Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩_{b} by Example 9.4 (1). We shall also use the notation ⟨⟨a_{1}, …, a_{n}⟩⟩ for the quadratic Pfister form ⟨⟨a_{1}, …, a_{n}]] in this case.

The class of an n-fold Pfister form (n > 0) belongs to

I_{q}^{n}(F) := I^{n−1}(F) · I_{q}(F).

As [a, b] = a[1, ab] for all a, b ∈ F with a ≠ 0, every nondegenerate binary quadratic form is a general 1-fold Pfister form. In particular, GP_{1}(F) generates I_{q}(F). It follows that GP_{n}(F) generates I_{q}^{n}(F) as an abelian group. In fact, as

(9.5)  a⟨⟨b, c]] = ⟨⟨ab, c]] − ⟨⟨a, c]]

for all a, b ∈ F^{×} and c ∈ F (with c ≠ 0 if char F ≠ 2), P_{n}(F) generates I_{q}^{n}(F) as an abelian group for n > 1.

Note that in the case char F ≠ 2, under the identification of I(F) with I_{q}(F), the group I^{n}(F) corresponds to I_{q}^{n}(F) and a bilinear Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩_{b} corresponds to the quadratic Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩.

Using the material in §98.E, we have the following example.

Example 9.6. Let A be a quaternion F -algebra.

(1) (Cf. Example 98.10.) Suppose that char F ≠ 2. If A = \binom{a,b}{F}, then the reduced quadratic norm form is equal to the quadratic form ⟨1, −a, −b, ab⟩ = ⟨⟨a, b⟩⟩.

(2) (Cf. Example 98.11.) Suppose that char F = 2. If A = \left[{a,b \atop F}\right], then the reduced quadratic norm form is equal to the quadratic form [1, ab] ⊥ [a, b]. This form is hyperbolic if a = 0 and is isomorphic to ⟨1, a⟩_{b} ⊗[1, ab] = ⟨⟨a, ab]] otherwise.

Example 9.7. Let L/F be a separable quadratic field extension and Q = (L/F, b), i.e., Q = L ⊕ Lj a quaternion F -algebra with j^{2} = b ∈ F^{×} (cf. §98.E). For any q = l +l′j ∈ Q, we have Nrd_{Q}(q) = N_{L}(l)−b N_{L}(l′). Therefore, Nrd_{Q} ≃ ⟨⟨b⟩⟩⊗N_{L}.

Proposition 9.8. Let φ be a round quadratic form and a ∈ F^{×}. Then (1) The form ⟨⟨a⟩⟩ ⊗ φ is also round.

(2) If φ is regular, then the following are equivalent: (i) ⟨⟨a⟩⟩ ⊗ φ is isotropic. (ii) ⟨⟨a⟩⟩ ⊗ φ is hyperbolic. (iii) a ∈ D(φ).

Proof. Set ψ = ⟨⟨a⟩⟩ ⊗ φ.

(1): Since 1 ∈ D(φ), it suffices to prove that D(ψ) ⊂ G(ψ). Let c be a nonzero value of ψ. Write c = x − ay for some x, y ∈ D̃(φ). If y = 0, we have c = x ∈ D(φ) = G(φ) ⊂ G(ψ). Similarly, y ∈ G(ψ) if x = 0, hence c = −ay ∈ G(ψ) as −a ∈ G(⟨⟨a⟩⟩) ⊂ G(ψ).

Now suppose that x and y are nonzero. Since φ is round, x, y ∈ G(φ) and, therefore,

ψ = φ ⊥ (−aφ) ≃ φ ⊥ (−ayx^{−1})φ = ⟨⟨ayx^{−1}⟩⟩ ⊗ φ.

By Example 1.14, we know that 1 − ayx^{−1} ∈ G(⟨⟨ayx^{−1}⟩⟩) ⊂ G(ψ). Since x ∈ G(φ) ⊂ G(ψ), we have c = (1 − ayx^{−1})x ∈ G(ψ).

(2): (i) ⇒ (iii): If φ is isotropic, then φ is universal by Proposition 7.13. So suppose that φ is anisotropic. Since ψ = φ ⊥ (−aφ) is isotropic, there exist x, y ∈ D(φ) such that x − ay = 0. Therefore a = xy^{−1} ∈ D(φ) as D(φ) is closed under multiplication.

(iii) ⇒ (ii): As φ is round, a ∈ D(φ) = G(φ) and ⟨⟨a⟩⟩ ⊗ φ is hyperbolic.

(ii) ⇒ (i) is trivial. □

Corollary 9.9. Quadratic Pfister forms are round.

Corollary 9.10. A quadratic Pfister form is either anisotropic or hyperbolic.

Proof. Suppose that ψ is an isotropic quadratic n-fold Pfister form. If n = 1, the result follows by Proposition 7.19(4). So assume that n > 1. Then ψ ≃ ⟨⟨a⟩⟩⊗φ for a quadratic Pfister form φ and the result follows by Proposition 9.8. □

Let char F = 2. We need another characterization of hyperbolic Pfister forms in this case. Let ℘ : F → F be the Artin-Schreier map defined by ℘(x) = x^{2} + x. (Cf. §98.B.) For a quadratic 1-fold Pfister form we have ⟨⟨d]] is hyperbolic if and only if d ∈ im(℘) by Example 98.3. More generally, we have:

Lemma 9.11. Let 𝔟 be an anisotropic bilinear Pfister form and d ∈ F. Then 𝔟 ⊗ ⟨⟨d]] is hyperbolic if and only if d ∈ im(℘) + D̃(𝔟′).

Proof. Suppose that 𝔟 ⊗ ⟨⟨d]] is hyperbolic and therefore isotropic. Let {e, f} be the standard basis of ⟨⟨d]]. Let v ⊗ e + w ⊗ f be an isotropic vector of 𝔟 ⊗ ⟨⟨d]] where v, w ∈ V_{𝔟}. We have a + b + cd = 0 where a = 𝔟(v, v), b = 𝔟(v, w), and c = 𝔟(w, w).

As 𝔟 is anisotropic, we have w ≠ 0, i.e., c ≠ 0. Suppose first that v = sw for some s ∈ F. Then 0 = a + b + cd = c(s^{2} + s + d), hence d = s^{2} + s ∈ im(℘).

Now suppose that v and w generate a 2-dimensional subspace W of V_{𝔟}. The determinant of 𝔟|_{W} is equal to xF^{×2} where x = b^{2} + bc + c^{2}d. Hence 𝔟|_{W} ≃ c⟨⟨x⟩⟩ by Example 1.10. As c ∈ D(𝔟) = G(𝔟) by Corollary 6.2, the form ⟨⟨x⟩⟩ is isometric to a subform of 𝔟. By the Bilinear Witt Cancellation (Corollary 1.28), we have ⟨x⟩ is a subform of 𝔟′, i.e., x ∈ D(𝔟′). Hence (b/c)^{2} + (b/c) + d = x/c^{2} ∈ D(𝔟′) and therefore d ∈ im(℘) + D(𝔟′).

Conversely, let d = x + y where x ∈ im(℘) and y ∈ D̃(𝔟′). If y = 0, then ⟨⟨d]] is hyperbolic, hence so is 𝔟 ⊗ ⟨⟨d]]. So suppose that y ≠ 0. By Lemma 6.11 there is a bilinear Pfister form 𝔠 such that 𝔟 ≃ 𝔠 ⊗ ⟨⟨y⟩⟩. Therefore, 𝔟 ⊗ ⟨⟨d]] ≃ 𝔠 ⊗ ⟨⟨y, d⟩⟩ is hyperbolic as ⟨⟨y, d]] ≃ ⟨⟨y, y]] by Example 98.3 which is hyperbolic. □

9.C. Annihilators. If φ is a nondegenerate quadratic form over F, then the annihilator of φ in W (F)

ann_{W (F)}(φ) := {𝔠 ∈ W (F) | 𝔠 · φ = 0}

is an ideal. When φ is a Pfister form this ideal has the structure that we had when φ was a bilinear anisotropic Pfister form. Indeed, the same proof yielding Proposition 6.22 and Corollary 6.23 shows:

Theorem 9.12. Let φ be anisotropic quadratic Pfister form. Then ann_{W (F)}(φ) is generated by binary symmetric bilinear forms ⟨⟨x⟩⟩_{𝔟} with x ∈ D(φ).

As in the bilinear case, if φ is 2-dimensional, we obtain stronger results. Indeed, the same proofs for the corresponding results show:

Lemma 9.13 (cf. Lemma 6.24). Let φ be a binary anisotropic quadratic form over F and 𝔠 an anisotropic bilinear form over F such that 𝔠 ⊗ φ is isotropic. Then 𝔠 ≃ 𝔡 ⊥ 𝔢 for some binary bilinear form 𝔡 annihilated by φ and bilinear form 𝔢 over F.

Proposition 9.14 (cf. Proposition 6.25). Let φ be a binary anisotropic quadratic form over F and 𝔠 an anisotropic bilinear form over F. Then there exist bilinear forms 𝔠_{1} and 𝔠_{2} over F such that 𝔠 ≃ 𝔠_{1} ⊥ 𝔠_{2} with 𝔠_{2} ⊗ φ anisotropic and 𝔠_{1} ≃ 𝔡_{1} ⊥ ⋯ ⊥ 𝔡_{n} where each 𝔡_{i} is a binary bilinear form annihilated by φ. In particular, − det 𝔡_{i} ∈ D(φ) for each i.

Corollary 9.15 (cf. Corollary 6.26). Let φ be a binary anisotropic quadratic form over F and 𝔠 an anisotropic bilinear form over F annihilated by 𝔟. Then 𝔠 ≃ 𝔡_{1} ⊥ ⋯ ⊥ 𝔡_{n} for some binary bilinear forms 𝔡_{i} annihilated by 𝔟 for i ∈ [1, n].

10. Totally singular forms

Totally singular forms in characteristic different from 2 are zero forms, but in characteristic 2 they become interesting. In this section, we look at totally singular forms in characteristic 2. In particular, throughout most of this section, char F = 2.

Let char F = 2. Let φ be a quadratic form over F. Then φ is a totally singular form if and only if it is diagonalizable. Moreover, if this is the case, then every basis of V_{φ} is orthogonal by Remark 7.24 and D̃(φ) is a vector space over the field F^{2}.

We investigate the F -subspace (D̃(φ))^{1/2} of F^{1/2}. Define an F -linear map

f : V_{φ} → (D̃(φ))^{1/2} given by f(v) = \sqrt{φ(v)}.

Then f is surjective and Ker(f) = rad φ. Let φ̃ be the quadratic form on (D̃(φ))^{1/2} over F defined by φ̃(\sqrt{a}) = a. Clearly φ̃ is anisotropic. Consequently, if φ̄ is the quadratic form induced on V_{φ}/ rad φ by φ, then f induces an isometry between φ̄ and φ̃. Moreover, φ̃ ≃ φ_{an}. Therefore, if char F = 2, the correspondence φ ↦ D̃(φ) gives rise to a bijection

Isometry classes of totally singular anisotropic quadratic forms

⟶^{∼}

Finite dimensional F^{2}-subspaces of F

Moreover, for any totally singular quadratic form φ, we have

dim φ_{an} = dim D̃(φ)

and if φ and ψ are two totally singular quadratic forms, then

φ ≃ ψ if and only if D(φ) = D(ψ) and dim φ = dim ψ.

We also have D̃(φ ⊥ ψ) = D̃(φ) + D̃(ψ).

Example 10.1. If F is a separably closed field of characteristic 2, the anisotropic quadratic forms are diagonalizable, hence totally singular.

Note that if 𝔟 is an alternating bilinear form and ψ is a totally singular quadratic form, then 𝔟⊗ψ = 0. It follows that the tensor product of totally singular quadratic forms φ ⊗ ψ := 𝔠 ⊗ ψ is well-defined where 𝔠 is a bilinear form with φ = φ_{𝔠}. The space D̃(φ ⊗ ψ) is spanned by D(φ) · D(ψ) over F^{2}.

Proposition 10.2. Let char F = 2. If φ is a totally singular quadratic form, then

G(φ) = {a ∈ F^{×} | aD(φ) ⊂ D(φ)}.

Proof. The inclusion “⊂” follows from Lemma 9.1. Conversely, let a ∈ F^{×} satisfy aD(φ) ⊂ D(φ). Then the F -linear map g : (D̃(φ))^{1/2} → (D̃(φ))^{1/2} defined by g(b) = \sqrt{a} b is an isometry between φ̃ and aφ̃. Therefore a ∈ G(φ̃) = G(φ). □

It follows from Proposition 10.2 that G̃(φ) := G(φ) ∪ {0} is a subfield of F containing F^{2} and D̃(φ) is a vector space over G̃(φ).

It is also convenient to introduce a variant of the notion of Pfister forms in all characteristics. A quadratic form φ is called a quasi-Pfister form if there exists a bilinear Pfister form 𝔟 with φ ≃ φ_{𝔟}, i.e., φ ≃ ⟨⟨a_{1}, …, a_{n}⟩⟩_{b} ⊗ ⟨1⟩_{q}. for some a_{1}, …, a_{n} ∈ F^{×}. Denote

⟨⟨a_{1}, …, a_{n}⟩⟩_{b} ⊗ ⟨1⟩_{q} by ⟨⟨a_{1}, …, a_{n}⟩⟩_{q}.

If char F ≠ 2, then the classes of quadratic Pfister and quasi-Pfister forms coincide. If char F = 2 every quasi-Pfister form is totally singular. Quasi-Pfister forms have some properties similar to those for quadratic Pfister forms.

Corollary 10.3. Quasi-Pfister forms are round.

Proof. Let 𝔟 be a bilinear Pfister form. As ⟨1⟩_{q} is a round quadratic form, the quadratic form 𝔟 ⊗ ⟨1⟩_{q} is round by Proposition 9.8. □

Remark 10.4. Let char F = 2. Let ρ = ⟨⟨a_{1}, …, a_{n}⟩⟩_{q} be an anisotropic quasi-Pfister form. Then D̃(ρ) is equal to the field F^{2}(a_{1}, …, a_{n}) of degree 2^{n} over F^{2}. Conversely, every field K such that F^{2} ⊂ K ⊂ F with [K : F^{2}] = 2^{n} is generated by n elements and therefore K = D̃(ρ) for an anisotropic n-fold quasi-Pfister form ρ. Thus we get a bijection

Isometry classes of anisotropic n-fold quasi-Pfister forms  ≃  Fields K with F^{2} ⊂ K ⊂ F and [K : F^{2}] = 2^{n}

Let φ be an anisotropic totally singular quadratic form. Then K = G̃(φ) is a field with K · D̃(φ) ⊂ D̃(φ). We have [K : F^{2}] < ∞ and D̃(φ) is a vector space over K. Let {b_{1}, …, b_{m}} be a basis of D̃(φ) over K and set ψ = ⟨b_{1}, …, b_{m}⟩_{q}. Choose an anisotropic n-fold quasi-Pfister form ρ such that D̃(ρ) = G̃(φ). As D̃(φ) is the vector space spanned by K · D(ψ) over F^{2}, we have φ ≃ ρ ⊗ ψ. In fact, ρ is the largest quasi-Pfister divisor of φ, i.e., quasi-Pfister form of maximal dimension such that φ ≃ ρ ⊗ ψ.

11. The Clifford algebra

To each quadratic form φ, one associates a ℤ/2ℤ-graded algebra by factoring the tensor algebra on V_{φ} by the relation φ(v) = v^{2}. This algebra, called the Clifford algebra generalizes the exterior algebra. In this section, we study the basic properties of Clifford algebras.

Let φ be a quadratic form on V over F. Define the Clifford algebra of φ to be the factor algebra C(φ) of the tensor algebra

T (V) := ∐_{n≥0} V^{⊗n}

modulo the ideal I generated by (v ⊗ v) − φ(v) for all v ∈ V. We shall view vectors in V as elements of C(φ) via the natural F -linear map V → C(φ). Note that v^{2} = φ(v) in C(φ) for every v ∈ V. The Clifford algebra of φ has a natural ℤ/2ℤ-grading

C(φ) = C_{0}(φ) ⊕ C_{1}(φ)

as I is homogeneous if degree is viewed modulo two. The subalgebra C_{0}(φ) is called the even Clifford algebra of φ. We have dim C(φ) = 2^{dim φ} and dim C_{0}(φ) = 2^{dim φ−1}. If K/F is a field extension, C(φ_{K}) = C(φ)_{K} and C_{0}(φ_{K}) = C_{0}(φ)_{K}.

Lemma 11.1. Let φ be a quadratic form on V over F with polar form 𝔟. Let v, w ∈ V. Then 𝔟(v, w) = vw + wv in C(φ). In particular, v and w are orthogonal if and only if vw = −wv in C(φ).

Proof. This follows from the polar identity. □

Example 11.2. (1) The Clifford algebra of the zero quadratic form on V coincides with the exterior algebra Λ(V).

(2) C_{0}(⟨a⟩) = F.

(3) If char F ≠ 2, then the Clifford algebra of the quadratic form ⟨a, b⟩ is C(⟨a, b⟩) = \binom{a,b}{F} and C_{0}(⟨a, b⟩) = F_{−ab}. In particular, C_{0}(⟨⟨b⟩⟩) = F_{b}.

(4) If char F = 2, then C([a, b]) = \left[{a,b \atop F}\right] and C_{0}([a, b]) = F_{ab}. In particular,

C_{0}(⟨⟨b]]) = F_{b}.

By construction, the Clifford algebra satisfies the following universal property: For any F -algebra A and any F -linear map f : V → A satisfying f(v)^{2} = φ(v) for all v ∈ V, there exists a unique F -algebra homomorphism f̃: C(φ) → A satisfying f̃(v) = f(v) for all v ∈ V.

Example 11.3. Let C(φ)^{op} denote the Clifford algebra of φ with the opposite multiplication. The canonical linear map V → C(φ)^{op} extends to an involution

: C(φ) → C(φ) given by the algebra isomorphism C(φ) → C(φ)^{op}. Note that if x = v_{1}v_{2} ⋯ v_{n}, then x̄ = v_{n} ⋯ v_{2}v_{1}.

Proposition 11.4. Let φ be a quadratic form on V over F and let a ∈ F^{×}. Then

(1) C_{0}(aφ) ≃ C_{0}(φ), i.e., the even Clifford algebras of similar quadratic forms are isomorphic.

(2) Let φ = ⟨a⟩ ⊥ ψ. Then C_{0}(φ) ≃ C(−aψ).

Proof. (1): Set K = F [t]/(t^{2} − a) = F ⊕ F t̄ with t̄ the image of t in K. Since (v⊗t̄)^{2} = φ(v)⊗t̄^{2} = aφ(v)⊗1 in C(φ)_{K} = C(φ)⊗_{F} K, there is an F -algebra homomorphism α : C(aφ) → C(φ)_{K} taking v ∈ V to v⊗t̄by the universal property of the Clifford algebra aφ. Since

(v ⊗ t̄)(v′ ⊗ t̄) = vv′ ⊗ t̄^{2} = avv′ ⊗ 1 ∈ C(φ) ⊂ C(φ)_{K},

the map α restricts to an F -algebra homomorphism C_{0}(aφ) → C_{0}(φ). As this map is clearly a surjective map of algebras of the same dimension, it is an isomorphism.

(2): Let V = F v ⊕ W with φ(v) = a and W ⊂ (F v)^{⊥}. Since

(vw)^{2} = −v^{2}w^{2} = −φ(v)ψ(w) = −aψ(w)

for every w ∈ W, the map W → C_{0}(φ) defined by w ↦ vw extends to an F -algebra isomorphism C(−aψ) →^{∼} C_{0}(φ) by the universal property of Clifford algebras. □

Let φ be a quadratic form on V over F. Applying the universal property of Clifford algebras to the natural linear map V → V/ rad φ → C(φ̄), where φ̄ is the induced quadratic form on V/ rad φ, we get a surjective F -algebra homomorphism C(φ) → C(φ̄) with kernel (rad φ)C(φ). Consequently, we get canonical isomorphisms

C(φ̄) ≃ C(φ)/(rad φ)C(φ),
C_{0}(φ̄) ≃ C_{0}(φ)/(rad φ)C_{1}(φ).

Example 11.5. Let φ = ℍ(W) be the hyperbolic form of the vector space V = W ⊕ W^{∗} with W a nonzero vector space. Then

C(φ) ≃ End_{F} (Λ(W)),

where the exterior algebra Λ(W) of V is considered as a vector space (cf. [86, Prop. 8.3]). Moreover,

C_{0}(φ) = End_{F} (Λ_{0}(W)) × End_{F} (Λ_{1}(W)),

where

Λ_{0}(W) := ⨁_{i≥0} Λ^{2i}(W) and Λ_{1}(W) := ⨁_{i≥0} Λ^{2i+1}(W).

In particular, C(φ) is a split central simple F -algebra and the center of C_{0}(φ) is the split quadratic étale F -algebra F × F. Note also that the natural F -linear map V → C(φ) is injective.

Proposition 11.6. Let φ be a quadratic form over F.

(1) If dim φ ≥ 2 is even, then the following conditions are equivalent: (a) φ is nondegenerate.

(b) C(φ) is central simple.

(c) C_{0}(φ) is separable with center Z(φ), a quadratic étale algebra.

(2) If dim φ ≥ 3 is odd, then the following conditions are equivalent: (a) φ is nondegenerate.

(b) C_{0}(φ) is central simple.

Proof. We may assume that F is algebraically closed. Suppose first that φ is nondegenerate and even-dimensional. Then φ is hyperbolic; and, by Example 11.5, the algebra C(φ) is a central simple F -algebra and C_{0}(φ) is a separable F -algebra whose center is the split quadratic étale F -algebra F × F.

Conversely, suppose that the even Clifford algebra C_{0}(φ) is separable or C(φ) is central simple. Let v ∈ rad φ. The ideals I = vC_{1}(φ) in C_{0}(φ) and J = vC(φ) in C(φ) satisfy I^{2} = 0 = J^{2}. Consequently, I = 0 or J = 0 as C_{0}(φ) is semi-simple or C(φ) is central simple and therefore v = 0 thus rad φ = 0. Thus φ is nondegenerate.

Now suppose that dim φ is odd. Write φ = ⟨a⟩ ⊥ ψ for some a ∈ F and an even-dimensional form ψ. Let v ∈ V_{φ} be a nonzero vector satisfying φ(v) = a with v orthogonal to V_{ψ}. If φ is nondegenerate, then a ≠ 0 and ψ is nondegenerate. It follows from Proposition 11.4(2) and the first part of the proof that the algebra C_{0}(φ) ≃ C(−aψ) is central simple.

Conversely, suppose that the algebra C_{0}(φ) is central simple. As dim φ ≥ 3, the subspace I := vC_{1}(φ) of C_{0}(φ) is nonzero. If a = 0, then I is a nontrivial ideal of C_{0}(φ), a contradiction to the simplicity of C_{0}(φ). Thus a ≠ 0 and by Proposition 11.4(2), C_{0}(φ) ≃ C(−aψ). Hence by the first part of the proof, the form ψ is nondegenerate. Therefore, φ is also nondegenerate. □

Lemma 11.7. Let φ be a nondegenerate quadratic form of positive even dimension. Then yx = x̄y for every x ∈ Z(φ) and y ∈ C_{1}(φ).

Proof. Let v ∈ V_{φ} be an anisotropic vector hence a unit in C(φ). Since conjugation by v on C(φ) stabilizes C_{0}(φ), it stabilizes the center of C_{0}(φ), i.e., vZ(φ)v^{−1} = Z(φ). As C(φ) is a central algebra, conjugation by v induces a nontrivial automorphism on Z(φ) given by x ↦ x̄ otherwise, since C_{1}(φ) = C_{0}(φ)v, the full algebra C(φ) would commute with Z(φ). Thus vx = x̄v for all x ∈ Z(φ). Let y ∈ C_{1}(φ). Writing y in the form y = zv for some z ∈ C_{0}(φ), we have yx = zvx = zx̄v = x̄zv = x̄y for every x ∈ Z(φ). □

Corollary 11.8. Let φ be a nondegenerate quadratic form of positive even dimension. If a is a norm for the quadratic étale algebra Z(φ) then C(aφ) ≃ C(φ).

Proof. Let x ∈ Z(φ) satisfy N(x) = a. By Lemma 11.7, we have (vx)^{2} = N(x)v^{2} = aφ(v) in C(φ) for every v ∈ V. By the universal property of the Clifford algebra aφ, there is an algebra homomorphism α : C(aφ) → C(φ) mapping v to vx. Since both algebras are simple of the same dimension, α is an isomorphism. □

12. Binary quadratic forms and quadratic algebras

In §98.B and §98.E we review the theory of quadratic and quaternion algebras. In this section, we study the relationship between these algebras and quadratic forms.

If A is a quadratic F -algebra, we let Tr_{A} and N_{A} denote the trace form and the quadratic norm form of A, respectively. (cf. §98.B). Note that N_{A} is a binary form representing 1.

Conversely, if φ is a binary quadratic form over F, then the even Clifford algebra C_{0}(φ) is a quadratic F -algebra. We have defined two maps

Quadratic ; F -algebras

\overrightarrow{⟵}

Binary quadratic forms representing 1

Proposition 12.1. The two maps above induce mutually inverse bijections of the set of isomorphism classes of quadratic F -algebras and the set of isometry classes of binary quadratic forms representing one. Under these bijections, we have:

(1) Quadratic étale algebras correspond to nondegenerate binary forms.

(2) Quadratic fields correspond to anisotropic binary forms.

(3) Semisimple algebras correspond to regular binary quadratic forms.

Proof. Let A be a quadratic F -algebra. We need to show that A ≃ C_{0}(N_{A}). We have C_{1}(N_{A}) = A. Therefore, the map α : A → C_{0}(N_{A}) defined by x ↦ 1 · x (where dot denotes the product in the Clifford algebra) is an F -linear isomorphism. We shall show that α is an algebra isomorphism, i.e., (1 · x) · (1 · y) = 1 · xy for all x, y ∈ A. The equality holds if x ∈ F or y ∈ F. Since A is 2-dimensional over F, it suffices to check the equality when x = y and it does not lie in F. We have 1 · x + x · 1 = N_{A}(x + 1) − N_{A}(x) − N_{A}(1) = Tr_{A}(x), so

(1 · x) · (1 · x) = (1 · x) · (Tr_{A}(x) − x · 1) = 1 · Tr_{A}(x)x − 1 · N_{A}(x) · 1 = 1 · x^{2}

as needed.

Conversely, let φ be a binary quadratic form on V representing 1. We shall show that the norm form for the quadratic F -algebra C_{0}(φ) is isometric to φ. Let v_{0} ∈ V be a vector satisfying φ(v_{0}) = 1. Let f : V → C_{0}(φ) be the F -linear isomorphism defined by f(v) = v · v_{0}. The quadratic equation (98.1) for v · v_{0} ∈ C_{0}(φ) in §98.B becomes

(v · v_{0})^{2} = v · (𝔟(v, v_{0}) − v · v_{0}) · v_{0} = 𝔟(v, v_{0})(v · v_{0}) − φ(v),

so N_{C_{0}(φ)}(v · v_{0}) = φ(v); hence

N_{C_{0}(φ)}(f(v)) = N_{C_{0}(φ)}(v · v_{0}) = φ(v),

i.e., f is an isometry of φ with the norm form of C_{0}(φ) as needed.

In order to prove that quadratic étale algebras correspond to nondegenerate binary forms it is sufficient to assume that F is algebraically closed. Then a quadratic étale algebra A is isomorphic to F × F and therefore N_{A} ≃ ℍ. Conversely, by Example 11.5, C_{0}(ℍ) ≃ F × F.

If a quadratic F -algebra A is a field, then obviously the norm form N_{A} is anisotropic. Conversely, if N_{A} is anisotropic, then for every nonzero a ∈ A we have aā = N_{A}(a) ≠ 0, therefore a is invertible, i.e., A is a field.

Statement (3) follows from statements (1) and (2), since a quadratic F -algebra is semisimple if and only if it is either a field or F × F, and a binary quadratic form is regular if and only if it is anisotropic or hyperbolic. □

Corollary 12.2. (1) Let A and B be quadratic F -algebras. Then A and B are isomorphic if and only if the norm forms N_{A} and N_{B} are isometric.

(2) Let φ and ψ be nonzero binary quadratic forms. Then φ and ψ are similar if and only if the even Clifford algebras C_{0}(φ) and C_{0}(ψ) are isomorphic.

Corollary 12.3. Let φ be an anisotropic binary quadratic form and let K/F be a quadratic field extension. Then φ_{K} is isotropic if and only if K ≃ C_{0}(φ).

Proof. By Proposition 12.1, the form φ_{K} is isotropic if and only if the 2dimensional even Clifford K-algebra C_{0}(φ_{K}) = C_{0}(φ) ⊗ K is not a field. The latter is equivalent to K ≃ C_{0}(φ). □

We now consider the relationship between quaternion and Clifford algebras.

Proposition 12.4. Let Q be a quaternion F -algebra and φ the reduced norm quadratic form of Q. Then C(φ) ≃ 𝕄_{2}(Q).

Proof. For every x ∈ Q, let m_{x} be the matrix \binom{0 x}{x̄ 0} in 𝕄_{2}(Q). Since m_{x}^{2} = xx̄ = Nrd(x) = φ(x), the F -linear map Q → 𝕄_{2}(Q) defined by x ↦ m_{x} extends to an F -algebra homomorphism α : C(φ) → 𝕄_{2}(Q) by the universal property of Clifford algebras. As C(φ) is a central simple algebra of dimension 16 = dim 𝕄_{2}(Q), the map α is an isomorphism. □

Corollary 12.5. Two quaternion algebras are isomorphic if and only if their reduced norm quadratic forms are isometric. In particular, a quaternion algebra is split if and only if its reduced norm quadratic form is hyperbolic.

Exercise 12.6. Let Q be a quaternion F -algebra and let φ′ be the restriction of the reduced norm quadratic form to the space Q′ of pure quaternions. Prove that φ′ is nondegenerate and C_{0}(φ′) is isomorphic to Q. Conversely, prove that any 3-dimensional nondegenerate quadratic form ψ is similar to the restriction of the reduced norm quadratic form of C_{0}(ψ) to the space of pure quaternions. Therefore, there is a natural bijection between the set of isomorphism classes of quaternion algebras over F and the set of similarity classes of 3-dimensional nondegenerate quadratic forms over F.

13. The discriminant

A major objective is to define sufficiently many invariants of quadratic forms. The first and simplest such invariant is the dimension. In this section, using quadratic étale algebras, we introduce a second invariant, the discriminant, of a nondegenerate quadratic form.

Let φ be a nondegenerate quadratic form over F of positive even dimension. The center Z(φ) of C_{0}(φ) is a quadratic étale F -algebra. The class of Z(φ) in Ét_{2}(F), the group of isomorphisms classes of quadratic étale F -algebras under the operation · induced by ⋆ (cf. §98.B), is called the discriminant of φ and will be denoted by disc(φ). Define the discriminant of the zero form to be trivial.

Example 13.1. By Example 11.2, we have disc(⟨a, b⟩) = F_{−ab} if char F ≠ 2 and disc([a, b]) = F_{ab} if char F = 2. It follows from Example 11.5 that the discriminant of a hyperbolic form is trivial.

The discriminant is a complete invariant for the similarity class of a nondegenerate binary quadratic form, i.e.,

Proposition 13.2. Two nondegenerate binary quadratic forms are similar if and only if their discriminants are equal.

Proof. Let disc(φ) = disc(ψ), i.e., C_{0}(φ) ≃ C_{0}(ψ). Write φ = aφ′ and ψ = bψ′, where φ′ and ψ′ both represent 1. By Proposition 12.1, the forms φ′ and ψ′ are the norm forms for C_{0}(φ′) = C_{0}(φ) and C_{0}(ψ′) = C_{0}(ψ), respectively. Since these algebras are isomorphic, we have φ′ ≃ ψ′. □

Corollary 13.3. A nondegenerate binary quadratic form φ is hyperbolic if and only if disc(φ) is trivial.

Lemma 13.4. Let φ and ψ be nondegenerate quadratic forms of even dimension over F. Then disc(φ ⊥ ψ) = disc(φ) · disc(ψ).

Proof. The even Clifford algebra C_{0}(φ ⊥ ψ) coincides with

(C_{0}(φ) ⊗_{F} C_{0}(ψ)) ⊕ (C_{1}(φ) ⊗_{F} C_{1}(ψ))

and contains Z(φ) ⊗_{F} Z(ψ). By Lemma 11.7, we have yx = x̄y for every x ∈ Z(φ) and y ∈ C_{1}(φ). Similarly, wz = z̄w for every z ∈ Z(ψ) and w ∈ C_{1}(ψ). Therefore, the center of C_{0}(φ ⊥ ψ) coincides with the subalgebra Z(φ) ⋆ Z(ψ) of all stable elements of Z(φ) ⊗_{F} Z(ψ) under the automorphism x ⊗ y ↦ x̄ ⊗ ȳ. □

Example 13.5. (1) Let char F ≠ 2. Then

disc(⟨a_{1}, a_{2}, …, a_{2n}⟩) = F_{c}

where c = (−1)^{n}a_{1}a_{2} … a_{2n}. For this reason, the discriminant is often called the signed determinant when the characteristic of F is different from 2.

(2) Let char F = 2. Then

disc([a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{n}, b_{n}]) = F_{c}

where c = a_{1}b_{1} + ⋯ + a_{n}b_{n}. The discriminant in the characteristic 2 case is often called the Arf invariant.

Proposition 13.6. Let ρ be a nondegenerate quadratic form over F. If disc(ρ) = 1 and ρ ⊥ ⟨a⟩ ∼ ⟨a⟩ for some a ∈ F^{×}, then ρ ∼ 0.

Proof. We may assume that the characteristic of F is 2. By Proposition 8.8, we have ρ ∼ [a, b] for some b ∈ F. Therefore, disc([a, b]) is trivial and [a, b] ∼ 0. □

It follows from Lemma 13.4 and Example 11.5 that the map

e_{1} : I_{q}(F) → Ét_{2}(F)

taking a form φ to disc(φ) is a well-defined group homomorphism. The analogue of Proposition 4.13 is true, viz.,

Theorem 13.7. The homomorphism e_{1} is surjective with kernel I_{q}^{2}(F).

Proof. The surjectivity follows from Example 13.1. Since similar forms have isomorphic even Clifford algebras, for any φ ∈ I_{q}(F) and a ∈ F^{×}, we have

e_{1}(⟨⟨a⟩⟩ · φ) = e_{1}(φ) · e_{1}(−aφ) = 0.

Therefore, e_{1}(I_{q}^{2}(F)) = 0.

Let φ ∈ I_{q}(F) be a form with trivial discriminant. We show by induction on dim φ that φ ∈ I_{q}^{2}(F). The case dim φ = 2 follows from Corollary 13.3. Suppose that dim φ ≥ 4. Write φ = ρ ⊥ ψ with ρ a binary form. Let a ∈ F^{×} be chosen so that the form φ′ = aρ ⊥ ψ is isotropic. Then the class of φ′ in I_{q}(F) is represented by a form of dimension less than dim φ. As disc(φ′) = disc(φ) is trivial, φ′ ∈ I_{q}^{2}(F) by induction. Since ρ ≡ aρ mod I_{q}^{2}(F), φ also lies in I_{q}^{2}(F). □

Remark 13.8. One can also define a discriminant like invariant for all nondegenerate quadratic forms. Let φ be a nondegenerate quadratic form. Define the determinant det φ of φ to be det 𝔟_{φ} in F^{×}/F^{×2} if the bilinear form 𝔟_{φ} is nondegenerate. If char F = 2 and dim φ is odd (the only remaining case), define det φ to be aF^{×2} in F^{×}/F^{×2} where a ∈ F^{×} satisfies φ|_{rad 𝔟_{φ}} ≃ ⟨a⟩.

Remark 13.9. Let φ be a nondegenerate even-dimensional quadratic form with trivial discriminant over F, i.e., φ ∈ I_{q}^{2}(F). Then Z(φ) ≃ F ×F, in particular, C(φ) is not a division algebra, i.e., C(φ) ≃ M_{2}(C^{+}(φ)) for a central simple F -algebra C^{+}(φ) uniquely determined up to isomorphism. Moreover, C_{0}(φ) ≃ C^{+}(φ) × C^{+}(φ).

14. The Clifford invariant

A more delicate invariant of a nondegenerate even-dimensional quadratic form arises from its associated Clifford algebra.

Let φ be a nondegenerate even-dimensional quadratic form over F. The Clifford algebra C(φ) is then a central simple F -algebra. Denote by clif(φ) the class of C(φ) in the Brauer group Br(F). It follows from Example 11.3 that clif(φ) ∈ Br_{2}(F). We call clif(φ) the Clifford invariant of φ.

Example 14.1. Let φ be the reduced norm form of a quaternion algebra Q. It follows from Proposition 12.4 that clif(φ) = [Q].

Lemma 14.2. Let φ and ψ be two nondegenerate even-dimensional quadratic forms over F. If disc(φ) is trivial, then clif(φ ⊥ ψ) = clif(φ) · clif(ψ).

Proof. Let e ∈ Z(φ) be a nontrivial idempotent and set s = e − ē = 1 − 2e. We have s̄ = −s, s^{2} = 1, and vs = s̄v = −sv for every v ∈ V_{φ} by Lemma 11.7. Therefore, in the Clifford algebra of φ ⊥ ψ, we have (v⊗1+s⊗w)^{2} = φ(v)+ψ(w) for all v ∈ V_{φ} and w ∈ V_{ψ}. It follows from the universal property of the Clifford algebra that the F -linear map V_{φ} ⊕ V_{ψ} → C(φ) ⊗_{F} C(ψ) defined by v ⊕ w ↦ v ⊗ 1 + s ⊗ w extends to an F -algebra homomorphism C(φ ⊥ ψ) → C(φ) ⊗_{F} C(ψ). This map is an isomorphism as the Clifford algebra of an even-dimensional form is central simple. □

Theorem 14.3. The map

e_{2} : I_{q}^{2}(F) → Br_{2}(F)

taking a form φ to clif(φ) is a well-defined group homomorphism. Moreover, I_{q}^{3}(F) ⊂ Ker(e_{2}).

Proof. It follows from Lemma 14.2 that e_{2} is well-defined. Next let φ ∈ I_{q}^{2}(F) and a ∈ F^{×}. Since disc(φ) is trivial, it follows from Corollary 11.8 that C(aφ) ≃ C(φ). Therefore, e_{2}(⟨⟨a⟩⟩ ⊗ φ) = e_{2}(φ) − e_{2}(aφ) = 0. □

We shall, in fact, prove that I_{q}^{3}(F) = Ker(e_{2}) in §16 for fields of characteristic 2 and Chapter VIII for fields of characteristic not 2.

15. Chain p-equivalence of quadratic Pfister forms

We saw that anisotropic bilinear Pfister forms ⟨⟨a_{1}, …, a_{n}⟩⟩ and ⟨⟨b_{1}, …, b_{n}⟩⟩ were p-chain equivalent if and only if they were isometric. This equivalence relation was based on isometries of 2-fold Pfister forms. In this section, we prove the analogous result for quadratic Pfister forms. This was first proven in the case of characteristic 2 by Aravire and Baeza in [12]. To begin we therefore need to establish isometries of quadratic 2-fold Pfister forms in characteristic 2. This is given by the following:

Lemma 15.1. Let F be a field of characteristic 2. Then in I_{q}(F), we have (1) ⟨⟨a, b + b′]] = ⟨⟨a, b]] + ⟨⟨a, b′]].

(2) ⟨⟨aa′, b]] ≡ ⟨⟨a, b]] + ⟨⟨a′, b]] mod I_{q}^{3}(F).

(3) ⟨⟨a + x^{2}, b]] = ⟨⟨a, \frac{ab}{a + x^{2}} ]].

(4) ⟨⟨a + a′, b]] ≡ ⟨⟨a, \frac{ab}{a + a′}]] + ⟨⟨a′, \frac{a′b}{a + a′}]] mod I_{q}^{3}(F).

Proof. (1): This follows by Example 7.23.

(2): Follows from the equality ⟨⟨a⟩⟩ + ⟨⟨a′⟩⟩ = ⟨⟨aa′⟩⟩ + ⟨⟨a, a′⟩⟩ in the Witt ring of bilinear forms by Example 4.10.

(3): Let c = b/(a + x^{2}) and

A = \left[{a, c \atop F}\right] and B = \left[{a + x^{2}, c \atop F}\right] .

By Corollary 12.5, it suffices to prove that A ≃ B. Let {1, i, j, ij} be the standard basis of A, i.e., i^{2} = a, j^{2} = b and ij + ji = 1. Considering the new basis {1, i + x, j, (i + x)j} with (i + x)^{2} = a + x^{2} shows that A ≃ B.

(4): We have by (1)–(3):

⟨⟨a + a′, b]] ≡ ⟨⟨\frac{a}{a′} + 1, b]] + ⟨⟨a′, b]] = ⟨⟨\frac{a}{a′} , \frac{ab}{a + a′}]] + ⟨⟨a′, b]]

≡ ⟨⟨a, \frac{ab}{a + a′}]] + ⟨⟨a′, \frac{ab}{a + a′}]] + ⟨⟨a′, b]]

= ⟨⟨a, \frac{ab}{a + a′}]] + ⟨⟨a′, \frac{a′b}{a + a′}]] mod I_{q}^{3}(F).  □

The definition of chain p-equivalence for quadratic Pfister forms is slightly more involved than that for bilinear Pfister forms.

Definition 15.2. Let a_{1}, …, a_{n−1}, b_{1}, …, b_{n−1} ∈ F^{×} and a_{n}, b_{n} ∈ F with n ≥ 1. We assume that a_{n} and b_{n} are nonzero if char F ≠ 2. Let φ = ⟨⟨a_{1}, …, a_{n−1}, a_{n}]] and ψ = ⟨⟨b_{1}, …, b_{n−1}, b_{n}]]. We say that the quadratic Pfister forms φ and ψ are

15. CHAIN p-EQUIVALENCE OF QUADRATIC PFISTER FORMS 65

simply p-equivalent if either n = 1 and ⟨⟨a_{1}]] ≃ ⟨⟨b_{1}]] or n ≥ 2 and there exist i and j with 1 ≤ i < j ≤ n satisfying

⟨⟨a_{i}, a_{j}⟩⟩ ≃ ⟨⟨b_{i}, b_{j}⟩⟩ with j < n and a_{l} = b_{l} for all l ≠ i, j or
⟨⟨a_{i}, a_{j}]] ≃ ⟨⟨b_{i}, b_{j}]] with j = n and a_{l} = b_{l} for all l ≠ i, j.

We say that φ and ψ are chain p-equivalent if there exist quadratic n-fold Pfister forms φ_{0}, …, φ_{m} for some m such that φ = φ_{0}, ψ = φ_{m} and φ_{i} is simply pequivalent to φ_{i+1} for each i ∈ [0, m − 1].

Theorem 15.3. Let

φ = ⟨⟨a_{1}, …, a_{n−1}, a_{n}]] and ψ = ⟨⟨b_{1}, …, b_{n−1}, b_{n}]]

be anisotropic quadratic n-fold Pfister forms. Then φ ≈ ψ if and only if φ ≃ ψ.

We shall prove this result in a series of steps. Suppose that φ ≃ ψ. The case char F ≠ 2 was considered in Theorem 6.10, so we may also assume that char F = 2. As before the map ℘ : F → F is defined by ℘(x) = x^{2} + x when char F = 2.

Lemma 15.4. Let char F = 2. If 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ is an anisotropic bilinear Pfister form and d_{1}, d_{2} ∈ F, then 𝔟 ⊗ ⟨⟨d_{1}]] ≃ 𝔟 ⊗ ⟨⟨d_{2}]] if and only if 𝔟 ⊗ ⟨⟨d_{1}]] ≈ 𝔟 ⊗ ⟨⟨d_{2}]].

Proof. Assume that 𝔟 ⊗ ⟨⟨d_{1}]] ≃ 𝔟 ⊗ ⟨⟨d_{2}]]. It follows from Example 7.23 that the form 𝔟 ⊗ ⟨⟨d_{1} + d_{2}]] is Witt-equivalent to 𝔟 ⊗ ⟨⟨d_{1}]] ⊥ 𝔟 ⊗ ⟨⟨d_{2}]] and hence is hyperbolic. By Lemma 9.11, we have d_{1} + d_{2} = x + y where x ∈ im(℘) and y ∈ D̃(𝔟′). If y = 0, then ⟨⟨d_{1}]] ≃ ⟨⟨d_{2}]] and we are done. So suppose that y ≠ 0. By Lemma 6.11, there is a bilinear Pfister form 𝔠 such that 𝔟 ≈ 𝔠 ⊗ ⟨⟨y⟩⟩. As ⟨⟨y, d_{1}]] ≃ ⟨⟨y, d_{2}]], we have

𝔟 ⊗ ⟨⟨d_{1}]] ≈ 𝔠 ⊗ ⟨⟨y, d_{1}]] ≈ 𝔠 ⊗ ⟨⟨y, d_{2}]] ≈ 𝔟 ⊗ ⟨⟨d_{2}]].  □

Lemma 15.5. Let char F = 2. If ρ = ⟨⟨b, b_{2}, …, b_{n}, d]] is a quadratic Pfister form, then for every a ∈ F^{×} and z ∈ D(ρ), we have ⟨⟨a⟩⟩ ⊗ ρ ≈ ⟨⟨az⟩⟩ ⊗ ρ.

Proof. We induct on dim ρ. Let η = ⟨⟨b_{2}, … b_{n}, d]]. We have z = x + by with x, y ∈ D̃(η). If y = 0, then x = z ≠ 0 and by the induction hypothesis ⟨⟨a⟩⟩ ⊗ η ≈ ⟨⟨az⟩⟩ ⊗ η, hence

⟨⟨a⟩⟩ ⊗ ρ = ⟨⟨a, b⟩⟩ ⊗ η ≈ ⟨⟨az, b⟩⟩ ⊗ η ≈ ⟨⟨az⟩⟩ ⊗ ρ.

If x = 0, then z = by and by the induction hypothesis ⟨⟨a⟩⟩ ⊗ η ≈ ⟨⟨ay⟩⟩ ⊗ η, hence

⟨⟨a⟩⟩ ⊗ ρ = ⟨⟨a, b⟩⟩ ⊗ η ≈ ⟨⟨ay, b⟩⟩ ⊗ η ≈ ⟨⟨az, b⟩⟩ ⊗ η ≈ ⟨⟨az⟩⟩ ⊗ ρ.

Now suppose that both x and y are nonzero. As η is round, xy ∈ D(η). By the induction hypothesis and Lemma 4.15,

⟨⟨a⟩⟩ ⊗ ρ = ⟨⟨a, b⟩⟩ ⊗ η ≈ ⟨⟨a, ab⟩⟩ ⊗ η ≈ ⟨⟨ax, aby⟩⟩ ⊗ η
≈ ⟨⟨az, bxy⟩⟩ ⊗ η ≈ ⟨⟨az, b⟩⟩ ⊗ η = ⟨⟨az⟩⟩ ⊗ ρ.  □

Lemma 15.6. Let char F = 2. Let 𝔟 = ⟨⟨a_{1}, …, a_{m}⟩⟩ be a bilinear Pfister form, ρ = ⟨⟨b_{1}, …, b_{n}, d]] a quadratic Pfister form with n ≥ 0 (if n = 0, then ρ = ⟨⟨d]]), and c ∈ F^{×}. Suppose there exists an x ∈ D(𝔟) with c+x ≠ 0 satisfying 𝔟⊗⟨⟨c+x⟩⟩⊗ρ is anisotropic. Then

𝔟 ⊗ ⟨⟨c + x⟩⟩ ⊗ ρ ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ψ

for some quadratic Pfister form ψ.

Proof. We proceed by induction on the dimension of 𝔟. Suppose 𝔟 = ⟨1⟩. Then x = y^{2} for some y ∈ F. It follows from Lemma 15.1 that ⟨⟨c + y^{2}, d]] ≃ ⟨⟨c, cd/(c + y^{2})]], hence ⟨⟨c + y^{2}⟩⟩ ⊗ ρ ≈ ⟨⟨c⟩⟩ ⊗ ⟨⟨b_{1}, …, b_{n}, cd/(c + y^{2})]].

So we may assume that dim 𝔟 > 1. Let 𝔠 = ⟨⟨a_{2}, …, a_{m}⟩⟩ and a = a_{1}. We have x = y + az where y, z ∈ D̃(𝔠). If c = az, then c + x = y belongs to D(𝔟), so the form 𝔟 ⊗ ⟨⟨c + x⟩⟩ would be metabolic contradicting hypothesis.

Let d := c + az. We have d ≠ 0. By the induction hypothesis,

𝔠 ⊗ ⟨⟨d + y⟩⟩ ⊗ ρ ≈ 𝔠 ⊗ ⟨⟨d⟩⟩ ⊗ μ and 𝔠 ⊗ ⟨⟨ac + a^{2}z⟩⟩ ⊗ μ ≈ 𝔠 ⊗ ⟨⟨ac⟩⟩ ⊗ ψ

for some quadratic Pfister forms μ and ψ. Hence by Lemma 4.15,

𝔟⊗⟨⟨c + x⟩⟩ ⊗ ρ = 𝔟 ⊗ ⟨⟨d + y⟩⟩ ⊗ ρ = 𝔠 ⊗ ⟨⟨a, d + y⟩⟩ ⊗ ρ
≈ 𝔠 ⊗ ⟨⟨a, d⟩⟩ ⊗ μ = 𝔠 ⊗ ⟨⟨a, c + az⟩⟩ ⊗ μ ≈ 𝔠 ⊗ ⟨⟨a, ac + a^{2}z⟩⟩ ⊗ μ
≈ 𝔠 ⊗ ⟨⟨a, ac⟩⟩ ⊗ ψ ≈ 𝔠 ⊗ ⟨⟨a, c⟩⟩ ⊗ ψ = 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ψ.  □

If 𝔟 is a bilinear Pfister form over a field F, then 𝔟 ≃ 𝔟′ ⊥ ⟨1⟩ with the pure subform 𝔟′ unique up to isometry. For a quadratic Pfister form over a field of characteristic 2, the analogue of this is not true. So, in this case, we have to modify our notion of a pure subform of a quadratic Pfister form. So suppose that char F = 2. Let φ = 𝔟 ⊗ ⟨⟨d]] be a quadratic Pfister form with 𝔟 = 𝔟′ ⊥ ⟨1⟩, a bilinear Pfister form. We have φ = ⟨⟨d]] ⊥ φ^{∘} with φ^{∘} = 𝔟′ ⊗ ⟨⟨d]]. The form φ^{∘} depends on the presentation of 𝔟. Let φ′ := ⟨1⟩ ⊥ 𝔟′ ⊗ ⟨⟨d]]. This form coincides with the complementary form ⟨1⟩^{⊥} in φ. The form φ′ is uniquely determined by φ up to isometry. Indeed, by Witt Extension Theorem 8.3, for any two vectors v, w ∈ V_{φ} with φ(v) = φ(w) = 1 there is an auto-isometry α of φ such that α(v) = w. Therefore, the orthogonal complements of F v and F w are isometric. We call the form φ′ the pure subform of φ.

Proposition 15.7. Let ρ = ⟨⟨b_{1}, …, b_{n}, d]] be a quadratic Pfister form, n ≥ 1, and 𝔟 = ⟨⟨a_{1}, …, a_{m}⟩⟩ a bilinear Pfister form. Set φ = 𝔟 ⊗ ρ. Suppose that φ is anisotropic. Let c ∈ D(𝔟 ⊗ ρ′) \ D(𝔟). Then φ ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ψ for some quadratic Pfister form ψ.

Proof. Let ρ = ⟨⟨b⟩⟩ ⊗ η with b = b_{1} and η = ⟨⟨b_{2}, …, b_{n}, d]] (with η = ⟨⟨d]] if n = 1). We proceed by induction on dim ρ. Note that if n = 1, we have η′ = ⟨1⟩ and D(𝔟 ⊗ η′) = D(𝔟). As

𝔟 ⊗ ρ′ = (𝔟 ⊗ η′) ⊥ (b𝔟 ⊗ η),

we have c = x + by with x ∈ D̃(𝔟 ⊗ η′), and y ∈ D̃(𝔟 ⊗ η).

If y = 0, then c = x ∈ D(𝔟 ⊗ η′) \ D(𝔟). In particular, n > 1. By the induction hypothesis, 𝔟 ⊗ η ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ μ for some quadratic Pfister form μ. Hence

φ = 𝔟 ⊗ ρ = 𝔟 ⊗ ⟨⟨b⟩⟩ ⊗ η ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ⟨⟨b⟩⟩ ⊗ μ.

Now suppose that y ≠ 0. By Lemma 15.5,

(15.8)  φ = 𝔟 ⊗ ρ = 𝔟 ⊗ ⟨⟨b⟩⟩ ⊗ η ≈ 𝔟 ⊗ ⟨⟨by⟩⟩ ⊗ η.

Assume that x ∉ D̃(𝔟). In particular, n > 1. By the induction hypothesis, 𝔟 ⊗ η ≈ 𝔟 ⊗ ⟨⟨x⟩⟩ ⊗ μ for some quadratic Pfister form μ. Therefore, by Lemma 4.15,

φ ≈ 𝔟 ⊗ ⟨⟨by⟩⟩ ⊗ η ≈ 𝔟 ⊗ ⟨⟨x, by⟩⟩ ⊗ μ ≈ 𝔟 ⊗ ⟨⟨c, bxy⟩⟩ ⊗ μ.

Finally, we assume that x ∈ D̃(𝔟). By Lemma 15.6 and (15.8),

φ ≈ 𝔟 ⊗ ⟨⟨by⟩⟩ ⊗ η = 𝔟 ⊗ ⟨⟨c + x⟩⟩ ⊗ η ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ψ

for a quadratic Pfister form ψ. □

Proof of Theorem 15.3. Let φ and ψ be isometric anisotropic quadratic n-fold Pfister forms over F as in the statement of Theorem 15.3. We must show that φ ≈ ψ. We may assume that char F = 2. Claim: For every r ∈ [0, n − 1], there exist a bilinear r-fold Pfister form 𝔟 and quadratic (n − r)-fold Pfister forms ρ and μ such that φ ≈ 𝔟 ⊗ ρ and ψ ≈ 𝔟 ⊗ μ.

We prove the claim by induction on r. The case r = 0 is obvious. Suppose we have such 𝔟, ρ and μ for some r < n − 1. Write ρ = ⟨⟨c⟩⟩ ⊗ ψ for some c ∈ F^{×} and quadratic Pfister form ψ, so φ ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ ψ. Note that as φ is anisotropic, we have c ∈ D(𝔟 ⊗ ρ′) \ D(𝔟).

The form 𝔟 ⊗ ⟨1⟩ is isometric to subforms of φ and ψ. As 𝔟_{φ} and 𝔟_{ψ} are nondegenerate, by the Witt Extension Theorem 8.3, an isometry between these subforms extends to an isometry between φ and ψ. This isometry induces an isometry of orthogonal complements 𝔟 ⊗ ρ′ and 𝔟 ⊗ μ′. Therefore, we have c ∈ D(𝔟 ⊗ ρ′) \ D(𝔟) = D(𝔟 ⊗ μ′) \ D(𝔟). It follows from Proposition 15.7 that ψ ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ σ for some quadratic Pfister form σ. Thus ψ ≈ 𝔟 ⊗ ⟨⟨c⟩⟩ ⊗ σ and the claim is established.

Applying the claim in the case r = n − 1, we find a bilinear (n − 1)-fold Pfister form 𝔟 and elements d_{1}, d_{2} ∈ F such that φ ≈ 𝔟 ⊗ ⟨⟨d_{1}]] and ψ ≈ 𝔟 ⊗ ⟨⟨d_{2}]]. By Lemma 15.4, we have 𝔟 ⊗ ⟨⟨d_{1}]] ≈ 𝔟 ⊗ ⟨⟨d_{2}]], hence φ ≈ ψ. □

16. Cohomological invariants

A major problem in the theory of quadratic forms was to determine the relationship between quadratic forms and Galois cohomology. In this section, using the cohomology groups defined in §101, we introduce the problem.

Let H^{n}(F) be the groups defined in §101. In particular,

H^{n}(F) ≃ \{ \begin{matrix}Ét_{2}(F), & if n = 1, \\ Br_{2}(F), & if n = 2. \end{matrix}

Let φ be a quadratic n-fold Pfister form. Suppose that φ ≃ ⟨⟨a_{1}, …, a_{n}]]. Define the cohomological invariant of φ to be the class e_{n}(φ) in H^{n}(F) given by

e_{n}(φ) = {a_{1}, a_{2}, …, a_{n−1}} · [F_{a_{n}}],

where [F_{c}] is the class of the étale quadratic extension F_{c}/F in Ét_{2}(F) ≃ H^{1}(F). The cohomological invariant e_{n} is well-defined on quadratic n-fold Pfister forms:

Proposition 16.1. Let φ and ψ be quadratic n-fold Pfister forms. If φ ≃ ψ, then e_{n}(φ) = e_{n}(ψ) in H^{n}(F).

Proof. This follows from Theorems 6.20 and 15.3. □

As in the bilinear case, if we use the Hauptsatz 23.7 below, we even have that if

φ ≡ ψ mod I_{q}^{n+1}(F), then e_{n}(φ) = e_{n}(ψ)

in H^{n}(F). (Cf. Corollary 23.9 below.) In fact, we shall also show by elementary means in Proposition 24.6 below that if φ_{1}, φ_{2} and φ_{3} are general quadratic n-fold Pfister forms such that φ_{1} + φ_{2} + φ_{3} ∈ I_{q}^{n+1}(F), then e_{n}(φ_{1}) + e_{n}(φ_{2}) + e_{n}(φ_{3}) = 0 ∈ H^{n}(F).

We call the extension of e_{n} to a group homomorphism e_{n} : I_{q}^{n}(F) → H^{n}(F) the nth cohomological invariant of I_{q}^{n}(F).

Fact 16.2. The nth cohomological invariant e_{n} exists for all fields F and for all n ≥ 1. Moreover, Ker(e_{n}) = I_{q}^{n+1}(F). Furthermore, there is a unique isomorphism

\overline{e}_{n} : I_{q}^{n}(F)/I_{q}^{n+1}(F) → H^{n}(F)

satisfying ē_{n}(φ + I_{q}^{n+1}(F)) = e_{n}(φ) for every quadratic n-fold Pfister quadratic form φ.

Special cases of Fact 16.2 can be proven by elementary methods. Indeed, we have already shown that the invariant e_{1} is well-defined on all of I_{q}(F) and coincides with the discriminant in Theorem 13.7 and e_{2} is well-defined on all of I_{q}^{2}(F) and coincides with the Clifford invariant by Theorem 14.3. Then by Theorems 13.7 and 14.3 the maps ē_{1} and ē_{2} are well-defined. For fields of characteristic different from 2, e_{3} was shown to be well-defined, hence ē_{3} by Arason in [4], and ē_{3} an isomorphism in [103], and independently by Rost [115]; and Jacob and Rost showed that e_{4} was well-defined in [65].

Suppose that char F ≠ 2. Then the identification of bilinear and quadratic forms leads to the composition

h_{F}^{n} : K_{n}(F)/2K_{n}(F) ⟶^{f_{n}} I^{n}(F)/I^{n+1}(F) = I_{q}^{n}(F)/I_{q}^{n+1}(F) ⟶^{\overline{e}_{n}} H^{n}(F),

where h_{F}^{n} is the norm residue homomorphism of degree n defined in §101.

Milnor conjectured that h_{F}^{n} was an isomorphism for all n in [106]. Voevodsky proved the Milnor Conjecture in [136]. As stated in Fact 5.15 the map f_{n} is an isomorphism for all n. In particular, e_{n} is well-defined and ē_{n} is an isomorphism for all n.

If char F = 2, Kato proved Fact 16.2 in [78].

We have proven that ē_{1} is an isomorphism in Theorem 13.7. We shall prove that h_{F}^{2} is an isomorphism in Chapter VIII below if the characteristic of F is different from 2. It follows that ē_{2} is an isomorphism. We now turn to the proof that ē_{2} is an isomorphism if char F = 2. The following statement was first proven by Sah in [118].

Theorem 16.3. Let char F = 2. Then ē_{2} : I_{q}^{2}(F)/I_{q}^{3}(F) → Br_{2}(F) is an isomorphism.

Proof. The classes of quaternion algebras generate the group Br_{2}(F) by [1, Ch. VII, Th. 30]. It follows that ē_{2} is surjective. So we need only show that ē_{2} is injective.

Let α ∈ I_{q}^{2}(F) satisfy e_{2}(α) = 1. Write α in the form ∑_{i=1}^{n} d_{i}⟨⟨a_{i}, b_{i}]]. By

assumption, the product of all \left[{a_{i}, c_{i} \atop F}\right] with c_{i} = b_{i}/a_{i} is trivial in Br(F).

We prove by induction on n that α ∈ I_{q}^{3}(F). If n = 1, we have α = ⟨⟨a_{1}, b_{1}]] and

e_{2}(α) = \left[{a_{1}, c_{1} \atop F}\right] = 1. Therefore, the reduced norm form α of the split quaternion

algebra \left[{a_{1}, c_{1} \atop F}\right] is hyperbolic by Corollary 12.5, hence α = 0.

In the general case, let L = F (a_{1}^{1/2}, …, a_{n−1}^{1/2}). The field L splits \left[{a_{i}, c_{i} \atop F}\right]

for all i ∈ [1, n − 1] and hence splits \left[{a_{n}, c_{n} \atop F}\right]. By Lemma 98.16, we have

\left[{a_{n}, c_{n} \atop F}\right] = \left[{c, d \atop F}\right], where c is the square of an element of L, i.e., c is the sum of

elements of the form g^{2}m where g ∈ F and m is a monomial in the a_{i}, i ∈ [1, n−1]. It follows from Corollary 12.5 that ⟨⟨a_{n}, b_{n}]] = ⟨⟨c, cd]]. By Lemma 15.1, we have ⟨⟨c, cd]] is congruent modulo I_{q}^{3}(F) to the sum of 2-fold Pfister forms ⟨⟨a_{i}, f_{i}]] with i ∈ [1, n − 1], f_{i} ∈ F. Therefore, we may assume that α = ∑_{i=1}^{n−1}⟨⟨a_{i}, b′_{i}]] for some b′_{i}. By the induction hypothesis, α ∈ I_{q}^{3}(F). □

CHAPTER III Forms over Rational Function Fields

17. The Cassels-Pfister Theorem

Given a quadratic form φ over a field over F, it is natural to consider values of the form over the rational function field F (t). The Cassels-Pfister Theorem shows that whenever φ represents a polynomial over F (t), then it already does so when viewed as a quadratic form over the polynomial ring F [t]. This results in specialization theorems. As an n-dimensional quadratic form ψ can be viewed as a polynomial in F [T] := F [t_{1}, …, t_{n}], one can also ask when is ψ(T) a value of φ_{F (T)}? If both the forms are anisotropic, we shall also show in this section the fundamental result that this is true if and only if ψ is a subform of φ.

Let φ be an anisotropic quadratic form on V over F and 𝔟 its polar form. Let v and u be two distinct vectors in V and set w = v − u. Let τ_{w} be the reflection with respect to w as defined in Example 7.2. Then φ(τ_{w}(v)) = φ(v) as τ_{w} is an isometry and

(17.1)  τ_{w}(v) = u + \frac{φ(u) − φ(v)}{φ(w)} w

as 𝔟_{φ}(v, w) = −𝔟_{φ}(v, −w) = −φ(u) + φ(v) + φ(w) by definition.

Notation 17.2. If T = (t_{1}, …, t_{n}) is a tuple of independent variables, let

F [T] := F [t_{1}, …, t_{n}] and F (T) := F (t_{1}, …, t_{n}).

If V is a finite dimensional vector space over F, let

V [T] := F [T] ⊗_{F} V and V (T) := V_{F (T)} := F (T) ⊗_{F} V.

Note that V (T) is also the localization of V [T] at F [T] \ {0}. In particular, if v ∈ V (T), there exist w ∈ V [T] and a nonzero f ∈ F [T] satisfying v = w/f. For a single variable t, we let V [t] := F [t] ⊗_{F} V and V (t) := V_{F (t)} := F (t) ⊗_{F} V.

The following general form of the Classical Cassels-Pfister Theorem is true.

Theorem 17.3 (Cassels-Pfister Theorem). Let φ be a quadratic form on V and let h ∈ F [t] ∩ D(φ_{F (t)}). Then there is a w ∈ V [t] such that φ(w) = h.

Proof. Suppose first that φ is anisotropic. Let v ∈ V (t) satisfy φ(v) = h. There is a nonzero polynomial f ∈ F [t] such that fv ∈ V [t]. Choose v and f so that deg(f) is the smallest possible. It suffice to show that f is constant. Suppose deg(f) > 0.

Using the analog of the Division Algorithm, we can divide the polynomial vector fv by f to get fv = fu + r, where u, r ∈ V [t] and deg(r) < deg(f). If r = 0, then v = u ∈ V [t] and f is constant; so we may assume that r ≠ 0. In particular, φ(r) ≠ 0 as φ is anisotropic. Set w = v − u = r/f and consider the vector

(17.4)  τ_{w}(v) = u + \frac{φ(u) − h}{φ(r)/f} r

as in (17.1). We have φ(τ_{w}(v)) = h. We show that f′ := φ(r)/f is a polynomial. As

f^{2}h = φ(fv) = φ(fu + r) = f^{2}φ(u) + f𝔟_{φ}(u, r) + φ(r),

we see that φ(r) is divisible by f. Equation (17.4) implies that f′τ_{w}(v) ∈ V [t] and the definition of r yields

deg(f′) = deg φ(r) − deg(f) < 2 deg(f) − deg(f) = deg(f),

a contradiction to the minimality of deg(f).

Now suppose that φ is isotropic. By Lemma 7.12, we may assume that rad φ = 0. In particular, a hyperbolic plane splits off as an orthogonal direct summand of φ by Lemma 7.13. Let e, e′ be a hyperbolic pair for this hyperbolic plane. Then

φ(he + e′) = 𝔟_{φ}(he, e′) = h𝔟_{φ}(e, e′) = h.  □

The theorem above was first proved by Cassels in [22] for the form ⟨1, …, 1⟩ over a field of characteristic not 2. This was generalized by Pfister to nondegenerate forms over such fields in [108] and used to prove the results through Corollary 17.13 below in that case.

Corollary 17.5. Let 𝔟 be a symmetric bilinear form on V and let h ∈ F [t] ∩ D(φ_{F (t)}). Then there is a v ∈ V [t] such that 𝔟(v, v) = h.

Proof. Let φ be φ_{𝔟}, i.e., φ(v) = 𝔟(v, v) for all v ∈ V. The result follows from the Cassels-Pfister Theorem for φ. □

Corollary 17.6. Let f ∈ F [t] be a sum of n squares in F (t). Then f is a sum of n squares in F [t].

Corollary 17.7 (Substitution Principle). Let φ be a quadratic form over F and h ∈ D(φ_{F (T)}) with T = (t_{1}, …, t_{n}). Suppose that h(x) is defined for x ∈ F^{n} and h(x) ≠ 0, then h(x) ∈ D(φ).

Proof. As h(x) is defined, we can write h = f/g with f, g ∈ F [T] and g(x) ≠ 0. Replacing h by g^{2}h, we may assume that h ∈ F [T]. Let T′ = (t_{1}, …, t_{n−1}) and x = (x_{1}, …, x_{n}). By the theorem, there exists v(T′, t_{n}) ∈ V (T′)[t_{n}] satisfying φ(v(T′, t_{n})) = h(T′, t_{n}). Evaluating t_{n} at x_{n} shows that h(T′, x_{n}) = φ(v(T′, x_{n})) ∈ D(φ_{F (T′)}). The conclusion follows by induction on n. □

As above, we also deduce:

Corollary 17.8 (Bilinear Substitution Principle). Let 𝔟 be a symmetric bilinear form over F and h ∈ D(𝔟_{F (T)}) with T = (t_{1}, …, t_{n}). Suppose that h(x) is defined for x ∈ F^{n} and h(x) ≠ 0, then h(x) ∈ D(𝔟).

We shall need the following slightly more general version of the Cassels-Pfister Theorem.

Proposition 17.9. Let φ be an anisotropic quadratic form on V. Suppose that s ∈ V and v ∈ V (t) satisfy φ(v) ∈ F [t] and 𝔟_{φ}(s, v) ∈ F [t]. Then there is w ∈ V [t] such that φ(w) = φ(v) and 𝔟_{φ}(s, w) = 𝔟_{φ}(s, v).

Proof. It suffices to show the value 𝔟_{φ}(s, v) does not change when v is modified in the course of the proof of Theorem 17.3. Choose v_{0} ∈ V [t] satisfying 𝔟_{φ}(s, v_{0}) = 𝔟_{φ}(s, v).

Let f ∈ F [t] be a nonzero polynomial such that fv ∈ V [t]. As the remainder r on dividing fv and fv − fv_{0} by f is the same and fv − fv_{0} ∈ (F (t)s)^{⊥}, we have r ∈ (F (t)s)^{⊥}. Therefore, 𝔟_{φ}(s, τ_{r}(v)) = 𝔟_{φ}(s, v). □

Lemma 17.10. Let φ be an anisotropic quadratic form and ρ a nondegenerate binary anisotropic quadratic form satisfying ρ(t_{1}, t_{2}) + d ∈ D(φ_{F (t_{1},t_{2})}) for some d ∈ F. Then φ ≃ ρ ⊥ μ for some form μ and d ∈ D̃(μ).

Proof. Let ρ(t_{1}, t_{2}) = at_{1}^{2} + bt_{1}t_{2} + ct_{2}^{2}. As ρ(t_{1}, t_{2}) + dt_{3}^{2} is a value of φ over F (t_{1}, t_{2}, t_{3}), there is a u ∈ V = V_{φ} such that φ(u) = a by the Substitution Principle 17.7. Applying the Cassels-Pfister Theorem 17.3 to the form φ_{F (t_{2})}, we find a v ∈ V_{F (t_{2})}[t_{1}] such that φ(v) = at_{1}^{2} + bt_{1}t_{2} + ct_{2}^{2} + d. Since φ is anisotropic, we have deg_{t_{1}} v ≤ 1, i.e., v(t_{1}) = v_{0} + v_{1}t_{1} for some v_{0}, v_{1} ∈ V_{F (t_{2})}. Expanding we get

φ(v_{0}) = a, 𝔟(v_{0}, v_{1}) = bt_{2}, φ(v_{1}) = ct_{2}^{2} + d,

where 𝔟 = 𝔟_{φ}. Clearly, v_{0} ∉ rad(𝔟_{F (t_{2})}).

We claim that u ∉ rad(𝔟). We may assume that u ≠ v_{0} and therefore

0 ≠ φ(u − v_{0}) = φ(u) + φ(v_{0}) − 𝔟(u, v_{0}) = 𝔟(u, u − v_{0})

as φ_{F (t_{2})} is anisotropic by Lemma 7.15, hence the claim follows.

By the Witt Extension Theorem 8.3, there is an isometry γ of φ_{F (t_{2})} satisfying γ(v_{0}) = u. Replacing v_{0} and v_{1} by u = γ(v_{0}) and γ(v_{1}) respectively, we may assume that v_{0} ∈ V.

Applying Proposition 17.9 to the vectors v_{0} and v_{1}, we find w ∈ V [t_{2}] satisfying φ(w) = ct_{2}^{2} + d and 𝔟(v_{0}, w) = bt_{2}. In a similar fashion, we have w = w_{0} + w_{1}t_{2} with w_{0}, w_{1} ∈ V. Expanding, we have

φ(v_{0}) = a, 𝔟(v_{0}, w_{1}) = b, φ(w_{1}) = c,
φ(w_{0}) = d, 𝔟(v_{0}, w_{0}) = 0, b(w_{0}, w_{1}) = 0.

It follows that if W is the subspace generated by v_{0} and w_{1}, then φ|_{W} ≃ ρ and d ∈ D̃(μ) where μ = φ|_{W^{⊥}}. □

Corollary 17.11. Let φ and ψ be two anisotropic quadratic forms over F with dim ψ = n. Let T = (t_{1}, …, t_{n}). Suppose that ψ(T) ∈ D(φ_{F (T)}). If ψ = ρ ⊥ σ with ρ a nondegenerate binary form and T′ = (t_{3}, …, t_{n}), then φ ≃ ρ ⊥ μ for some form μ and μ(T′) ∈ D̃_{F (T′)}(φ_{F (T′)}).

Theorem 17.12 (Representation Theorem). Let φ and ψ be two anisotropic quadratic forms over F with dim ψ = n. Let T = (t_{1}, …, t_{n}). Then the following are equivalent:

(1) D(ψ_{K}) ⊂ D(φ_{K}) for every field extension K/F.

(2) ψ(T) ∈ D(φ_{F (T)}).

(3) ψ is isometric to a subform of φ.

In particular, if any of the above conditions hold, then dim ψ ≤ dim φ.

Proof. (1) ⇒ (2) and (3) ⇒ (1) are trivial.

(2) ⇒ (3): Applying the structure results, Propositions 7.31 and 7.29, we can write ψ = ψ_{1} ⊥ ψ_{2}, where ψ_{1} is an orthogonal sum of nondegenerate binary forms and ψ_{2} is diagonalizable. Repeated application of Corollary 17.11 allows us to reduce to the case ψ = ψ_{2}, i.e., ψ = ⟨a_{1}, …, a_{n}⟩ is diagonalizable.

We induct on n. The case n = 1 follows from the Substitution Principle 17.7. Suppose that n = 2. Then we have a_{1}t^{2} + a_{2} ∈ D(φ_{F (t)}). By the Cassels-Pfister Theorem, there is a v ∈ V [t] where V = V_{φ} satisfying φ(v) = a_{1}t^{2} + a_{2}. As φ is anisotropic, we have v = v_{1} + v_{2}t for v_{1}, v_{2} ∈ V and therefore φ(v_{1}) = a_{1}, φ(v_{2}) = a_{2}, and 𝔟(v_{1}, v_{2}) = 0. The restriction of φ on the subspace spanned by v_{1} and v_{2} is isometric to ψ.

In the general case, set T = (t_{1}, t_{2}, …, t_{n}), T′ = (t_{2}, …, t_{n}), b = a_{2}t^{2} + ⋯ + a_{n}t_{n}^{2}. As a_{1}t^{2} + b is a value of φ over F (T′)(t), by the case considered above, there are vectors v_{1}, v_{2} ∈ V_{F (T′)} satisfying

φ(v_{1}) = a_{1}, φ(v_{2}) = b and 𝔟(v_{1}, v_{2}) = 0.

It follows from the Substitution Principle 17.7 that there is w ∈ V such that φ(w) = a_{1}.

We claim that there is an isometry γ of φ over F (T′) such that φ(v_{1}) = w. We may assume that w ≠ v_{1} as φ_{F (T′)} is anisotropic by Lemma 7.15. We have

0 ≠ φ(w − v_{1}) = φ(w) + φ(v_{1}) − 𝔟(w, v_{1}) = 𝔟(w, w − v_{1}) = 𝔟(v_{1} − w, v_{1}),

therefore w and v_{1} do not belong to rad 𝔟. The claim follows by the Witt Extension Theorem 8.3.

Replacing v_{1} and v_{2} by γ(v_{1}) = w and γ(v_{2}) respectively, we may assume that v_{1} ∈ V. Set W = (F v_{1})^{⊥}. Note that v_{2} ∈ W_{F (T′)}, hence b is a value of φ|_{W} over F (T′). By the induction hypothesis applied to the forms ψ′ = ⟨a_{2}, …, a_{n}⟩ and φ|_{W}, there is a subspace V′ ⊂ W such that φ|_{V′} ≃ ⟨a_{2}, …, a_{n}⟩. Note that v_{1} is orthogonal to V′ and v_{1} ∉ V′ as ψ is anisotropic. Therefore, the restriction of φ on the subspace F v_{1} ⊕ V′ is isometric to ψ. □

A field F is called formally real if −1 is not a sum of squares. In particular, char F = 0 if this is the case (cf. §95).

Corollary 17.13. Suppose that F is formally real and T = (t_{0}, …, t_{n}). Then t_{0}^{2} + t_{1}^{2} + ⋯ + t_{n}^{2} is not a sum of n squares in F (T).

Proof. If this is false, then t_{0}^{2} + t_{1}^{2} + ⋯ + t_{n}^{2} ∈ D(n⟨1⟩). As (n + 1)⟨1⟩ is anisotropic, this contradicts the Representation Theorem. □

The ideas above also allow us to develop a test for simultaneous zeros for quadratic forms. This was proven independently by Amer in [2] and Brumer in [21] for fields of characteristic not 2 and by Leep for arbitrary fields in [92].

Theorem 17.14. Let φ and ψ be two quadratic forms on a vector space V over F. Then the form φ_{F (t)} + tψ_{F (t)} on V (t) over F (t) is isotropic if and only if φ and ψ have a common isotropic vector in V.

Proof. Clearly, a common isotropic vector for φ and ψ is also an isotropic vector for ρ := φ_{F (t)} + tψ_{F (t)}.

Conversely, let ρ be isotropic. There exists a nonzero v ∈ V [t] such that ρ(v) = 0. Choose such a v of the smallest degree. We claim that deg v = 0, i.e., v ∈ V. If we show this, the equality φ(v) + tψ(v) = 0 implies that v is a common isotropic vector for φ and ψ.

Suppose n := deg v > 0. Write v = w + t^{n}u with u ∈ V and w ∈ V [t] of degree less than n. Note that by assumption ρ(u) ≠ 0. Consider the vector

v′ = ρ(u) · τ_{u}(v) = ρ(u)v − 𝔟_{ρ}(v, u)u ∈ V [t].

As ρ(v) = 0, we have ρ(v′) = 0. It follows from the equality

ρ(w)v − 𝔟_{ρ}(v, w)w = ρ(v − t^{n}u)v − 𝔟_{ρ}(v, v − t^{n}u)(v − t^{n}u) = t^{2n}(ρ(u)v − 𝔟_{ρ}(v, u)u)

that

v′ = \frac{ρ(w)v − 𝔟_{ρ}(v, w)w}{t^{2n}} .

Note that deg ρ(w) ≤ 2n − 1 and deg 𝔟_{ρ}(v, w) ≤ 2n. Therefore, deg v′ < n, contradicting the minimality of n. □

18. Values of forms

Let φ be an anisotropic quadratic form over F. Let p ∈ F [T] = F [t_{1}, …, t_{n}] be irreducible and F (p) the quotient field of F [T]/(p). In this section, we determine what it means for φ_{F (p)} to be isotropic. We base our presentation on ideas of Knebusch in [82]. This result has consequences for finite extensions K/F. In particular, the classical Springer’s Theorem that forms remain anisotropic under odd degree extensions follows as well as a norm principle about values of φ_{K}.

Order the group ℤ^{n} lexicographically, i.e., (i_{1}, …, i_{n}) < (j_{1}, …, j_{n}) if for the first integer k satisfying i_{k} ≠ j_{k} with 1 ≤ k ≤ n, we have i_{k} < j_{k}. If i = (i_{1}, …, i_{n}) in ℤ^{n} and a ∈ F^{×}, write aT^{i} for at_{1}^{i_{1}} ⋯ t_{n}^{i_{n}} and call i the degree of aT^{i}. Let f = aT^{i} + monomials of lower degree in F [T] with a ∈ F^{×}. The term aT^{i} is called the leading term of f. We define the degree deg f of f to be i, the degree of the leading term, and the leading coefficient f^{∗} of f to be a, the coefficient of the leading term. Let T_{f} denote T^{i} if i is the degree of the leading term of f. Then f = f^{∗}T_{f} + f′ with deg f′ < deg T_{f}. For convenience, we view deg 0 < deg f for every nonzero f ∈ F [T]. Note that deg(fg) = deg f + deg g and (fg)^{∗} = f^{∗}g^{∗}. If h ∈ F (T)^{×} and h = f/g with f, g ∈ F [T], let h^{∗} = f^{∗}/g^{∗}.

Let V be a finite dimensional vector space over F. For every nonzero v ∈ V [T] define the degree deg v, the leading vector v^{∗}, and the leading term v^{∗}T_{v} in a similar fashion. Let deg 0 < deg v for any nonzero v ∈ V [T]. So if v ∈ V [T] is nonzero, we have v = v^{∗}T_{v} + v′ with deg v′ < deg T_{v}.

Lemma 18.1. Let φ be a quadratic form on V over F and g ∈ F [T]. Suppose that g ∈ D(φ_{F (T)}). Then g^{∗} ∈ D(φ). If, in addition, φ is anisotropic, then deg g ∈ 2ℤ^{n}.

Proof. Since φ on V and the induced quadratic form φ̄ on V/ rad φ have the same values, we may assume that rad(φ) = 0. In particular, if φ is isotropic, it is universal, so we may assume that φ anisotropic.

Let g = φ(v) with v ∈ V (T). Write v = w/f with w ∈ V [T] and nonzero f ∈ F [T]. Then f^{2}g = φ(w). As (f^{2}g)^{∗} = (f^{∗})^{2}g^{∗}, we may assume that v ∈ F [T]. Let v = v^{∗}T_{v} + v′ with deg v′ < deg v. Then

g = φ(v^{∗}T_{v}) + 𝔟_{φ}(v^{∗}T_{v}, v′) + φ(v′) = φ(v^{∗})T_{v}^{2} + 𝔟_{φ}(v^{∗}, v′)T_{v} + φ(v′)
= φ(v^{∗})T_{v}^{2} + terms of lower degree.

As φ is anisotropic, we must have φ(v^{∗}) ≠ 0, hence g^{∗} = φ(v^{∗}) ∈ D(φ). As the leading term of g is φ(v^{∗})T_{v}^{2}, the second statement also follows. □

Let v ∈ V [T]. Suppose that f ∈ F [T] satisfies deg_{t_{1}} f > 0. Let T′ = (t_{2}, …, t_{n}). Viewing v ∈ V (T′)[t_{1}], the analog of the usual division algorithm produces an equation

v = fw′ + r′ with w′, r′ ∈ V_{F (T′)}[t_{1}] and deg_{t_{1}} r′ < deg_{t_{1}} f.

Clearing denominators in F [T′], we get hv = fw + r

with w, r ∈ V [T], 0 ≠h ∈ F [T′] and deg_{t_{1}} r < deg_{t_{1}} f,
so deg h < deg f, deg r < deg f.

(18.2)

If φ is a quadratic form over F let ⟨D(φ)⟩ denote the subgroup in F^{×} generated by D(φ).

Theorem 18.3 (Quadratic Value Theorem). Let φ be an arbitrary anisotropic quadratic form on V and f ∈ F [T] a nonzero polynomial. Then the following conditions are equivalent:

(1) f^{∗}f ∈ ⟨D(φ_{F (T)})⟩.

(2) There exists an a ∈ F^{×} such that af ∈ ⟨D(φ_{F (T)})⟩.

(3) φ_{F (p)} is isotropic for each irreducible divisor p occurring to an odd power in the factorization of f.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3): Let af ∈ ⟨D(φ_{F (T)})⟩, i.e., there are

0 ≠ h ∈ F [T] and v_{1}, …, v_{m} ∈ V [T]

such that ah^{2}f = ∏ φ(v_{i}). Let p be an irreducible divisor of f to an odd power. Write v_{i} = p^{k_{i}}v′_{i} so that v′_{i} is not divisible by p. Dividing out both sides by p^{2k}, with k = ∑ k_{i}, we see that the product ∏ φ(v′_{i}) is divisible by p. Hence the residue of one of the φ(v′_{i}) is trivial in the residue field F (p) while the residue of v′_{i} is not trivial. Therefore, f_{F (p)} is isotropic.

(3) ⇒ (1): We proceed by induction on n and deg f. The statement is obvious if f = f^{∗}. In the general case, we may assume that f is irreducible. Therefore, by assumption φ_{F (f)} is isotropic. In particular, we see that there exists a vector v ∈ V_{φ}[T] such that f | φ(v) and f |̸ v. If deg_{t_{1}} f = 0, let T′ = (t_{2}, …, t_{n}) and let L denote the quotient field of F [T′]/(f). Then F (f) = L(t_{1}) so φ_{L} is isotropic by Lemma 7.15 and we are done by induction on n. Therefore, we may assume that deg_{t_{1}} f > 0. By (18.2), there exist 0 ≠ h ∈ F [T] and w, r ∈ V [T] such that hv = fw + r with deg h < deg f and deg r < deg f. As

φ(hv) = φ(fw + r) = f^{2}φ(w) + f𝔟_{φ}(w, r) + φ(r),

we have f | φ(r). If r = 0, then f | hv. But f is irreducible and f |̸ v, so f | h. This is impossible as deg h < deg f. Thus r ≠ 0. Let φ(r) = fg for some g ∈ F [T]. As φ is anisotropic g ≠ 0. So we have fg ∈ D(φ_{F (T)}), hence also (fg)^{∗} = f^{∗}g^{∗} ∈ D(φ) by Lemma 18.1.

Let p be an irreducible divisor occurring to an odd power in the factorization of g. As deg φ(r) < 2 deg f, we have deg g < deg f, hence p occurs with the same multiplicity in the factorization of fg. By (2) ⇒ (3), applied to the polynomial fg, the form φ_{F (p)} is isotropic. Hence the induction hypothesis implies that g^{∗}g ∈ ⟨D(φ_{F (T)})⟩. Consequently, f^{∗}f = f^{∗}^{2} · (f^{∗}g^{∗})^{−1} · g^{∗}g · fg · g^{−2} ∈ ⟨D(φ_{F (T)})⟩. □

Theorem 18.4 (Bilinear Value Theorem). Let 𝔟 be an anisotropic symmetric bilinear form on V and f ∈ F [T] a nonzero polynomial. Then the following conditions are equivalent:

(1) f^{∗}f ∈ ⟨D(𝔟_{F (T)})⟩.

(2) There exists an a ∈ F^{×} such that af ∈ ⟨D(𝔟_{F (T)})⟩.

(3) 𝔟_{F (p)} is isotropic for each irreducible divisor p occurring to an odd power in the factorization of f.

Proof. Let φ = φ_{𝔟}. As D(𝔟_{K}) = D(φ_{K}) for every field extension K/F by Lemma 9.2 and 𝔟_{K} is isotropic if and only if φ_{K} is isotropic, the result follows by the Quadratic Value Theorem 18.3. □

A basic result in Artin-Schreier theory is that an ordering on a formally real field extends to an ordering on a finite algebraic extension of odd degree, equivalently if the bilinear form n⟨1⟩ is anisotropic over F for any integer n, it remains so over any finite extension of odd degree. Witt conjectured in [139] that any anisotropic symmetric bilinear form remains anisotropic under an odd degree extension (if char F ≠ 2). This was first shown to be true by Springer in [126]. This is in fact true without a characteristic assumption for both quadratic and symmetric bilinear forms.

Corollary 18.5 (Springer’s Theorem). Let K/F be a finite extension of odd degree. Suppose that φ (respectively, 𝔟) is an anisotropic quadratic form (respectively, symmetric bilinear form) over F. Then φ_{K} (respectively, 𝔟_{K}) is anisotropic.

Proof. By induction on [K : F], we may assume that K = F (θ) is a primitive extension. Let p be the minimal polynomial of θ over F. Suppose that φ_{K} is isotropic. Then ap ∈ ⟨D(φ_{F (t)})⟩ for some a ∈ F^{×} by the Quadratic Value Theorem 18.3. It follows that p has even degree by Lemma 18.1, a contradiction. If 𝔟 is a symmetric bilinear form over F, applying the above to the quadratic form φ_{𝔟} shows the theorem also holds in the bilinear case. □

We shall give another proof of Springer’s Theorem in Corollary 71.3 below.

Corollary 18.6. If K/F is an extension of odd degree, then r_{K/F} : W (F) → W (K) and r_{K/F} : I_{q}(F) → I_{q}(K) are injective.

Corollary 18.7. Let φ and ψ be two quadratic forms on a vector space V over F having no common isotropic vector in V. Then for any field extension K/F of odd degree the forms φ_{K} and ψ_{K} have no common isotropic vector in V_{K}.

Proof. This follows from Springer’s Theorem and Theorem 17.14. □

Exercise 18.8. Let char F ≠ 2 and K/F be a finite purely inseparable field extension. Then r_{K/F} : W (F) → W (K) is an isomorphism.

Corollary 18.9. Let K = F (θ) be an algebraic extension of F and p the (monic) minimal polynomial of θ over F. Let φ be a regular quadratic form over F. Suppose that there exists a c ∈ F such that p(c) ∉ ⟨D(φ)⟩. Then φ_{K} is anisotropic.

Proof. As rad φ = 0, if φ were isotropic it would be universal. Thus φ is anisotropic. In particular, p is not linear, hence p(c) ≠ 0. Suppose that φ_{K} is isotropic. By the Quadratic Value Theorem 18.3, we have p ∈ ⟨D(φ_{F (t)})⟩. By the Substitution Principle 17.7, we have p(c) ∈ ⟨D(φ)⟩ for all c ∈ F, a contradiction.

□

As another consequence, we obtain the following theorem first proved by Knebusch in [81].

Theorem 18.10 (Value Norm Principle). Let φ be a quadratic form over F and K/F a finite field extension. Then N_{K/F} (D(φ_{K})) ⊂ ⟨D(φ)⟩.

Proof. Let V = V_{φ}. Since the form φ on V and the induced form φ̄ on V/ rad(φ) have the same values, we may assume that rad(φ) = 0. If φ is isotropic, then φ splits off a hyperbolic plane. In particular, φ is universal and the statement is obvious. Thus we may assume that φ is anisotropic. Moreover, we may assume that dim φ ≥ 2 and 1 ∈ D(φ).

Case 1: φ_{K} is isotropic.

Let x ∈ D(φ_{K}). Suppose that K = F (x). Let p ∈ F [t] denote the (monic) minimal polynomial of x so K = F (p). It follows from the Quadratic Value Theorem 18.3 that p ∈ ⟨D(φ_{F (t)})⟩ and deg p is even. In particular, N_{K/F} (x) = p(0) and by the Substitution Principle 17.7,

N_{K/F} (x) = p(0) ∈ ⟨D(φ)⟩.

If F (x) ⊊ K, let m = [K : F (x)]. If m is even, then N_{K/F} (x) ∈ F^{×2} ⊂ ⟨D(φ)⟩. If m is odd, then φ_{F (x)} is isotropic by Springer’s Theorem 18.5. Applying the above argument to the field extension F (x)/F yields

N_{K/F} (x) = N_{F (x)/F} (x)^{m} ∈ ⟨D(φ)⟩

as needed.

Case 2: φ_{K} is anisotropic.

Let x ∈ D(φ_{K}). Choose vectors v, v_{0} ∈ V_{K} such that φ_{K}(v) = x and φ_{K}(v_{0}) = 1. Let V′ ⊂ V_{K} be a 2-dimensional subspace (over K) containing v and v_{0}. The restriction φ′ of φ_{K} to V′ is a binary anisotropic quadratic form over K representing x and 1. It follows from Proposition 12.1 that the even Clifford algebra L = C_{0}(φ′) is a quadratic field extension of K and x = N_{L/K}(y) for some y ∈ L^{×}. Moreover, since C_{0}(φ′_{L}) = C_{0}(φ′) ⊗_{K} L = L ⊗_{K} L is not a field, by the same proposition, φ′ and therefore φ is isotropic over L. Applying Case 1 to the field extension L/F yields

N_{K/F} (x) = N_{K/F} (N_{L/K}(y)) = N_{L/F} (y) ∈ ⟨D(φ)⟩.  □

Theorem 18.11 (Bilinear Value Norm Principle). Let 𝔟 be a symmetric bilinear form over F and let K/F be a finite field extension. Then N_{K/F} (D(𝔟_{K})) ⊂ ⟨D(𝔟)⟩.

Proof. As D(𝔟_{E}) = D(φ_{𝔟_{E}}) for any field extension E/F, this follows from the quadratic version of the theorem. □

19. Forms over a discrete valuation ring

We wish to look at similarity factors of bilinear and quadratic forms. To do so we need a few facts about such forms over a discrete valuation ring (DVR) which we now establish. These results are based on the work of Springer in [127] in the case of fields of characteristic different from 2.

Throughout this section, R will be a DVR with quotient field K, residue field K̄, and prime element π. If V is a free R-module of finite rank, then the definition of a (symmetric) bilinear form and quadratic form on V is analogous to the field case. In particular, we can associate to every quadratic form its polar form 𝔟_{φ} : (v, w) ↦ φ(v + w) − φ(v) − φ(w). Orthogonal complements are defined in the usual way. Orthogonal sums of bilinear (respectively, quadratic) forms are defined as in the field case. We use analogous notation as in the field case when clear. If F → R is a ring homomorphism and φ is a quadratic form over F, we let φ_{R} = R ⊗_{F} φ.

A bilinear form 𝔟 on V is nondegenerate if l : V → Hom_{R}(V, R) defined by v ↦ l_{v} : w → b(v, w) is an isomorphism. As in the field case, we have the crucial

Proposition 19.1. Let R be a DVR. Let V be a free R-module of finite rank and W a submodule of V. If φ is a quadratic form on V with 𝔟_{φ}|_{W} nondegenerate, then φ = φ|_{W} ⊥ φ|_{W^{⊥}}.

Proof. As 𝔟_{φ}|_{W} is nondegenerate, W ∩ W^{⊥} = {0} and if v ∈ V, there exists w′ ∈ W such that the linear map W → F by w ↦ 𝔟_{φ}(v, w) is given by 𝔟_{φ}(v, w) = 𝔟_{φ}(w′, w) for all w ∈ W. Consequently, v = w + (v − w′) ∈ W ⊕ W^{⊥} and the result follows. □

Hyperbolic quadratic forms and planes are also defined in an analogous way. We let ℍ denote the quadratic hyperbolic plane.

If R is a DVR and V a vector space over the quotient field K of R. A vector v ∈ V is called primitive if it is not divisible by a prime element π, i.e., the image v̄ of v in K̄ ⊗_{R} V is not zero.

Arguing as in Proposition 7.13, we have

Lemma 19.2. Let R be a DVR. Let φ be a quadratic form on V whose polar form is nondegenerate. Suppose that V contains an isotropic vector v. Then there exists a submodule W of V containing v such that φ|_{W} ≃ ℍ.

Proof. Dividing v by π^{n} for an appropriate choice of n, we may assume that v is primitive. It follows easily that V/Rv is torsion-free, hence free. In particular, V → V/Rv splits, therefore, Rv is a direct summand of V. Let f : V → R be an R-linear map satisfying f(v) = 1. As l : V → Hom_{R}(V, R) is an isomorphism, there exists an element w ∈ V such that f = l_{w}, hence 𝔟_{φ}(v, w) = 1. Let W = Rv ⊕ Rw. Then v, w − φ(w)v is a hyperbolic pair. □

By induction, we conclude:

Corollary 19.3. Suppose that R is a DVR. Let φ be a quadratic form on V over R whose polar form is nondegenerate. Then φ = φ|_{V_{1}} ⊥ φ|_{V_{2}} with V_{1}, V_{2} submodules of V satisfying φ|_{V_{1}} is anisotropic and φ|_{V_{2}} ≃ mℍ for some m ≥ 0.

Associated to a quadratic form φ on V over R are two forms: φ_{K} on K ⊗_{R} V over K and φ̄ = φK̄ on K̄ ⊗_{R} V over K̄.

Lemma 19.4. Suppose that R is a complete DVR. Let φ be an anisotropic quadratic form over R whose associated bilinear form 𝔟_{φ} is nondegenerate. Then φ̄ is also anisotropic.

Proof. Let {v_{1}, …, v_{n}} be a basis for V_{φ} and t_{1}, …, t_{n} the respective coordi-

nates. If w ∈ V_{φ}, then \frac{∂φ}{∂t_{i}} (w) = b_{φ}(v_{i}, w). In particular, if w̄ ≠ ¯0, there exists an i

such that b̄_{φ}(v̄_{i}, w̄) ≠ 0. It follows by Hensel’s lemma that φ would be isotropic if φ̄ is. □

Lemma 19.5. Let φ and ψ be two quadratic forms over a DVR R such that φ̄ and ψ̄ are anisotropic over K̄. Then φ_{K} ⊥ πψ_{K} is anisotropic over K.

Proof. Suppose that φ(u) + πψ(v) = 0 for some u ∈ V_{φ} and v ∈ V_{ψ} with at least one of u and v primitive. Reducing modulo π, we have φ̄(ū) = 0. Since φ̄ is anisotropic, u = πw for some w. Therefore, πφ(w) + ψ(v) = 0 and reducing modulo π we get ψ̄(v̄) = 0. Since ψ̄ is also anisotropic, v is divisible by π, a contradiction. □

Corollary 19.6. Let φ and ψ be anisotropic forms over F. Then φ_{F (t)} ⊥ tψ_{F (t)} is anisotropic.

Proof. In the lemma, let R = F [t]_{(t)}, a DVR, π = t a prime. As \overline{φ_{R}} = φ and \overline{ψ_{R}} = ψ, the result follows from the lemma. □

Proposition 19.7. Let φ be a quadratic form over a complete DVR R such that the associated bilinear form 𝔟_{φ} is nondegenerate. Suppose that φ_{K} ≃ πφ_{K}. Then φ̄ is hyperbolic.

Proof. Write φ = ψ ⊥ nℍ with ψ anisotropic. By Lemma 19.4, we have ψ̄ is anisotropic. The form

φ_{K} ⊥ (−πφ_{K}) ≃ ψ_{K} ⊥ (−πψ_{K}) ⊥ 2nℍ

is hyperbolic and ψ_{K} ⊥ (−πψ_{K}) is anisotropic over K by Lemma 19.5. We must have ψ = 0 by uniqueness of the Witt decomposition over K, hence φ = nℍ is hyperbolic. It follows that φ̄ is hyperbolic. □

Proposition 19.8. Let φ be a nondegenerate quadratic form over F of even dimension. Let f ∈ F [T] and p ∈ F [T] an irreducible polynomial factor of f of odd multiplicity. If φ_{F (T)} ≃ fφ_{F (T)}, then φ_{F (p)} is hyperbolic.

Proof. Let R denote the completion of the DVR F [T]_{(p)} and let K be its quotient field. The residue field of R coincides with F (p). Modifying f by a square, we may assume that f = up for some u ∈ R^{×}. As φ_{F (T)} ≃ fφ_{F (T)}, we have φ_{F (T)} ≃ upφ_{F (T)}. Applying Proposition 19.7 to the form φ_{R} and π = up yields \overline{(φ_{R})} = φ_{F (p)} is hyperbolic. □

We shall also need the following:

Proposition 19.9. Let R be a DVR with quotient field K. Let φ and ψ be two quadratic forms on V and W over R, respectively, such that their respective residue forms φ̄ and ψ̄ are anisotropic. If φ_{K} ≃ ψ_{K}, then φ ≃ ψ (over R).

Proof. Let f : V_{K} → W_{K} be an isometry between φ_{K} and ψ_{K}. It suffices to prove that f(V) ⊂ W and f^{−1}(W) ⊂ V. Suppose that there exists a v ∈ V such that f(v) is not in W. Then f(v) = w/π^{k} for some primitive w ∈ W and k > 0. Since f is an isometry, we have ψ(w) = π^{2k}φ(v), i.e., ψ(w) is divisible by π, hence w̄ is an isotropic vector of ψ̄, a contradiction. Analogously, f^{−1}(W) ⊂ V. □

19.A. Residue homomorphisms. If R is a DVR, then for each x ∈ K^{×} we can write x = uπ^{n} for some u ∈ R^{×} and n ∈ ℤ.

Lemma 19.10. Let R be a DVR with quotient field K and residue field K̄. Let π be a prime element in R. There exist group homomorphisms

∂ : W (K) → W (K̄) and ∂_{π} : W (K) → W (K̄)

satisfying

∂(⟨uπ^{n}⟩) = \begin{cases}⟨ū⟩ & n is even \\ 0 & n is odd \end{cases} and ∂_{π}(⟨uπ^{n}⟩) = \begin{cases}0 & n is even, \\ ⟨ū⟩ & n is odd, \end{cases}

for u ∈ R^{×} and n ∈ ℤ.

Proof. It suffices to prove the existence of ∂ as we can take ∂_{π} = ∂ ∘λ_{π} where λ_{π} is the group homomorphism λ_{π} : W (K) → W (K) given by 𝔟 → π𝔟.

By Theorem 4.8, it suffices to check that the generating relations of the Witt ring are respected. As ⟨\overline{1}⟩ + ⟨\overline{−1}⟩ = 0 in W (K̄), it suffices to show if a, b ∈ R with a + b ≠ 0, then

(19.11)  ∂(⟨a⟩) + ∂(⟨b⟩) = ∂(⟨a + b⟩) + ∂(⟨ab(a + b)⟩)

in W (K̄).

Let

a = a_{0}π^{n}, b = b_{0}π^{m}, a + b = π^{l}c_{0} with a_{0}, b_{0}, c_{0} ∈ R^{×},

and m, n, l ∈ ℤ satisfying min{m, n} ≤ l. We may assume that n ≤ m. Suppose that n < m. Then

a + b = π^{n}a_{0}(1 + π^{m−n} \frac{b_{0}}{a_{0}} ) and ab(a + b) = π^{2n+m}b_{0}a_{0}^{2}(1 + \frac{b_{0}}{a_{0}} π^{m−n}).

In particular, ∂(⟨a⟩) = ∂(⟨a + b⟩) and ∂(⟨b⟩) = ∂(⟨ab(a + b)⟩) as needed.

Suppose that n = m. If n = l, then a_{0} + b_{0} ∈ R^{×} and the result follows by the Witt relation in W (K̄).

So suppose that n < l. Then ā_{0} = −b̄_{0} so the left hand side of (19.11) is zero. If l is odd, then ∂(⟨a + b⟩) = 0 = ∂(⟨ab(a + b)⟩) as needed. So we may assume that l is even. Then ⟨a + b⟩ ≃ ⟨c_{0}⟩ and ⟨ab(a + b)⟩ ≃ ⟨a_{0}b_{0}c_{0}⟩ over K. Hence the right hand side of (19.11) is ⟨c̄_{0}⟩ + ⟨ā_{0}b̄_{0}c̄_{0}⟩ = ⟨c̄_{0}⟩ + ⟨−c̄_{0}⟩ = 0 in W (K̄) also. □

The map ∂ : W (K) → W (K̄) in the lemma is not dependent on the choice on the prime element π. It is called the first residue homomorphism with respect to R. The map ∂_{π} : W (K) → W (K̄) does depend on π. It is called the second residue homomorphism with respect to R and π.

Remark 19.12. Let R be a DVR with quotient field K and residue field K̄. Let π be a prime element in R. If 𝔟 is a nondegenerate diagonalizable bilinear form over K, we can write 𝔟 as

𝔟 ≃ ⟨u_{1}, …, u_{n}⟩ ⊥ π⟨v_{1}, …, v_{m}⟩

for some u_{i}, v_{j} ∈ R^{×}. Then ∂(𝔟) = ⟨ū_{1}, …, ū_{n}⟩ in W (K̄) and ∂_{π}(𝔟) = ⟨v̄_{1}, …, v̄_{m}⟩ in W (K̄).

Example 19.13. Let R be a DVR with quotient field K and residue field K̄. Let π be a prime element in R. Let 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ be an anisotropic n-fold Pfister form over K. Then we may assume that a_{i} = π^{j_{i}}u_{i} with j_{i} = 0 or 1 and u_{i} ∈ R^{×} for all i. By Corollary 6.13, we may assume that a_{i} ∈ R^{×} for all i > 1. As 𝔟 = −a_{1}⟨⟨a_{2}, …, a_{n}⟩⟩ ⊥ ⟨⟨a_{2}, …, a_{n}⟩⟩, if a_{1} ∈ R^{×}, then ∂(𝔟) = ⟨⟨ā_{1}, …, ā_{n}⟩⟩ and ∂_{π}(𝔟) = 0, and if a_{1} = πu_{1}, then ∂(𝔟) = ⟨⟨ā_{2}, …, ā_{n}⟩⟩ and ∂_{π}(𝔟) = −ū_{1}⟨⟨ā_{2}, …, ā_{n}⟩⟩.

As n-fold Pfister forms generate I^{n}(F), we have, by the example, the following:

Lemma 19.14. Let R be a DVR with quotient field K and residue field K̄. Let π be a prime element in R. Then for every n ≥ 1:

(1) ∂(I^{n}(K)) ⊂ I^{n−1}(K̄).

(2) ∂_{π}(I^{n}(K)) ⊂ I^{n−1}(K̄).

Exercise 19.15. Suppose that R is a complete DVR with quotient field K and residue field K̄. If char K̄ ≠ 2, then the residue homomorphisms induce split exact sequences of groups:

0 → W (K̄) → W (K) → W (K̄) → 0

and

0 → I^{n}(K̄) → I^{n}(K) → I^{n−1}(K̄) → 0.

20. Similarities of forms

Let φ be an anisotropic quadratic form over F. Let p ∈ F [T] := F [t_{1}, …, t_{n}] be irreducible and F (p) the quotient field of F [T]/(p). In this section, we determine what it means for φ_{F (p)} to be hyperbolic. We also establish the analogous result for anisotropic bilinear forms over F. We saw that a form becoming isotropic over F (p) was related to the values it represented over the polynomial ring F [T]. We shall see that hyperbolicity is related to the similarity factors of the form over F [T]. We shall also deduce norm principles for similarity factors of a form over F first established by Scharlau in [119] and Knebusch in [82]. To establish these results, we introduce the transfer of forms from a finite extension of F to F, an idea introduced by Scharlau (cf. [119]) for quadratic forms over fields of characteristic different from 2 and Baeza (cf. [15]) in arbitrary characteristic.

20.A. Transfer of bilinear and quadratic forms. Let K/F be a finite field extension and s : K → F an F -linear functional. If 𝔟 is a symmetric bilinear form on V over K define the transfer s_{∗}(𝔟) of 𝔟 induced by s to be the symmetric bilinear form on V over F given by

s_{∗}(𝔟)(v, w) = s(𝔟(v, w)) for all v, w ∈ V.

If φ is a quadratic form on V over K, define the transfer s_{∗}(φ) of φ induced by s to be the quadratic form on V over F given by s_{∗}(φ)(v) = s(φ(v)) for all v ∈ V with polar form s_{∗}(𝔟_{φ}).

Note that dim s_{∗}(𝔟) = [K : F] dim 𝔟.

Lemma 20.1. Let K/F be a finite field extension and s : K → F an F -linear functional. The transfer s_{∗} factors through orthogonal sums and preserves isometries.

Proof. Let v, w ∈ V_{𝔟}. If 𝔟(v, w) = 0, then s_{∗}(𝔟)(v, w) = s(𝔟(v, w)) = 0. Thus s_{∗}(𝔟 ⊥ 𝔠) = s_{∗}(𝔟) ⊥ s_{∗}(𝔠). If σ : 𝔟 → 𝔟′ is an isometry, then

s_{∗}(𝔟′)(σ(v), σ(w)) = s(𝔟′(σ(v), σ(w))) = s(𝔟(v, w)) = s_{∗}(𝔟)(v, w),

so σ : s_{∗}(𝔟) → s_{∗}(𝔟′) is also an isometry. □

Proposition 20.2 (Frobenius Reciprocity). Suppose that K/F is a finite extension of fields and s : K → F an F -linear functional. Let 𝔟 and 𝔠 be symmetric bilinear forms over F and K, respectively, and φ and ψ quadratic forms over F and K respectively. Then there exist canonical isometries:

s_{∗}(𝔟_{K} ⊗_{K} 𝔠) ≃ 𝔟 ⊗_{F} s_{∗}(𝔠), (20.3a)

s_{∗}(𝔟_{K} ⊗_{K} ψ) ≃ 𝔟 ⊗_{F} s_{∗}(ψ), (20.3b)

s_{∗}(𝔠 ⊗_{K} φ_{K}) ≃ s_{∗}(𝔠) ⊗_{F} φ. (20.3c)

In particular,

s_{∗}(𝔟_{K}) ≃ 𝔟 ⊗_{F} s_{∗}(⟨1⟩_{b}).

Proof. (a): The canonical F -linear map V_{𝔟_{K}} ⊗_{K} V_{𝔠} → V_{𝔟} ⊗_{F} V_{𝔠} given by (a ⊗ v) ⊗ w ↦ v ⊗ aw is an isometry. Indeed,

s(𝔟_{K} ⊗ 𝔠)((a ⊗ v) ⊗ w, (a′ ⊗ v′) ⊗ w′) = s(aa′𝔟(v, v′)𝔠(w, w′))
= 𝔟(v, v′)s(𝔠(aw, a′w′))
= (𝔟 ⊗ s𝔠)(v ⊗ aw, v′ ⊗ a′w′).

The last statement follows from the first by setting 𝔠 = ⟨1⟩.

(b) and (c) are proved in a similar fashion. □

Lemma 20.4. Let K/F be a finite field extension and s : K → F a nonzero F - linear functional.

(1) If 𝔟 is a nondegenerate symmetric bilinear form on V over K, then s_{∗}(𝔟) is nondegenerate on V over F.

(2) If φ is an even-dimensional nondegenerate quadratic form on V over K, then s_{∗}(φ) is nondegenerate on V over F.

Proof. Suppose that 0 ≠ v ∈ V. As 𝔟 is nondegenerate, there exists a w ∈ V such that 1 = 𝔟(v, w). As s is not zero, there exists a c ∈ K such that 0 ≠ s(c) = s_{∗}(𝔟)(v, cw). This shows (1). Statement (2) follows from (1) and Remark 7.21(1). □

Corollary 20.5. Let K/F be a finite extension of fields and s : K → F a nonzero F -linear functional.

(1) If 𝔠 is a bilinear hyperbolic form over K, then s_{∗}(𝔠) is a hyperbolic form over F.

(2) If φ is a quadratic hyperbolic form over K, then s_{∗}(φ) is a hyperbolic form over F.

Proof. (1): As s_{∗} respects orthogonality, we may assume that 𝔠 = ℍ_{1}. By Frobenius Reciprocity,

s_{∗}(ℍ_{1}) ≃ s_{∗}((ℍ_{1})_{K}) ≃ (ℍ_{1})_{F} ⊗ s_{∗}(⟨1⟩).

As s_{∗}(⟨1⟩) is nondegenerate by Lemma 20.4, we have s_{∗}(ℍ_{1}) is hyperbolic by Lemma 2.1.

(2): This follows in the same way as (1) using Lemma 8.16. □

Let K/F be a finite field extension and s : K → F a nonzero F -linear functional. By Lemmas 20.4 and 20.5, the functional s induces group homomorphisms

s_{∗} : Ŵ(K) → Ŵ(F), s_{∗} : W (K) → W (F), and s_{∗} : I_{q}(K) → I_{q}(F)

called transfer maps. Let 𝔟 and 𝔠 be nondegenerate symmetric bilinear forms over F and K, respectively, and φ and ψ nondegenerate quadratic forms over F and K, respectively. By Frobenius Reciprocity, we have

s_{∗}(r_{K/F} (𝔟) · 𝔠) = 𝔟 · s_{∗}(𝔠)

in Ŵ(F) and W (F), i.e., s_{∗} : Ŵ(K) → Ŵ(F) is a Ŵ(F)-module homomorphism and s_{∗} : W (K) → W (F) is a W (F)-module homomorphism where we view W (K) as a W (F)-module via r_{K/F}. Furthermore,

s_{∗}(r_{K/F} (𝔟) · ψ) = 𝔟 · s_{∗}(ψ) and s_{∗}(𝔠 · r_{K/F} (φ)) = s_{∗}(𝔠) · φ

in I_{q}(F). Note that s_{∗}(I(K)) ⊂ I(F).

Corollary 20.6. Let K/F be a finite field extension and s : K → F a nonzero F -linear functional. Then the compositions

s_{∗}r_{K/F} : Ŵ(F) → Ŵ(F), s_{∗}r_{K/F} : W (F) → W (F), and s_{∗}r_{K/F} : I_{q}(F) → I_{q}(F)

are given by multiplication by s_{∗}(⟨1⟩_{b}), i.e., 𝔟 ↦ 𝔟 · s_{∗}(⟨1⟩_{b}) for a nondegenerate symmetric bilinear form 𝔟 and φ ↦ s_{∗}(⟨1⟩_{b}) · φ for a nondegenerate quadratic form φ.

Corollary 20.7. Let K/F be a field extension and s : K → F a nonzero F - linear functional. Then im(s_{∗}) is an ideal in Ŵ(F) (respectively, W (F)) and is independent of s.

Proof. By Frobenius Reciprocity, im(s_{∗}) is an ideal. Suppose that s_{1} : K → F is another nonzero F -linear functional. Let K → Hom_{F} (K, F) be the F - isomorphism given by a ↦ (x ↦ s(ax)). Hence there exists a unique a ∈ K^{×} such that s_{1}(x) = s(ax) for all x ∈ K. Consequently, (s_{1})_{∗}(𝔟) = s_{∗}(a𝔟) for all nondegenerate symmetric bilinear forms 𝔟 over K. □

Let K = F (x)/F be an extension of degree n and a = N_{K/F} (x) ∈ F^{×} the norm of x. Let

s : K → F be the F -linear functional defined by

s(1) = 1 and s(x^{i}) = 0 for all i ∈ [1, n − 1].

(20.8)

Then s(x^{n}) = (−1)^{n+1}a.

Lemma 20.9. The transfer induced by the F -linear functional s in (20.8) satisfies

s_{∗}(⟨1⟩_{b}) = \{ \begin{matrix}⟨1⟩_{b} & if n is odd, \\ ⟨1, −a⟩_{b} & if n is even. \end{matrix}

Proof. Let 𝔟 = s_{∗}(⟨1⟩). Let V ⊂ K be the F -subspace spanned by x^{i} with i ∈ [1, n], a nondegenerate subspace. Then V^{⊥} = F, consequently K = F ⊕ V.

First suppose that n = 2m + 1 is odd. The subspace of W spanned by x^{i}, i ∈ [1, m], is a lagrangian of 𝔟|_{V}, hence 𝔟|_{V} is metabolic and 𝔟 = 𝔟|_{V^{⊥}} = ⟨1⟩ in W (F).

Next suppose that n = 2m is even. We have

𝔟(x^{i}, x^{j}) = \{ \begin{matrix}0 & if i + j < n, \\ −a & if i + j = n. \end{matrix}

It follows that det 𝔟 = (−1)^{m}aF^{×2} and the subspace W′ ⊂ W spanned by all x^{i} with i ≠ m and 1 ≤ i ≤ n is nondegenerate. In particular, K = W′ ⊕ (W′)^{⊥} by Proposition 1.6. By dimension count dim(W′)^{⊥} = 2. As the subspace of W′ spanned by x^{i}, i ∈ [1, m − 1], is a lagrangian of 𝔟|_{W′}, we have 𝔟|_{W′} is metabolic. Computing determinants, yields 𝔟|_{(W′)^{⊥}} ≃ ⟨1, −a⟩, hence in W (F), we have 𝔟 = 𝔟|_{(W′)^{⊥}} = ⟨1, −a⟩. □

Corollary 20.10. Suppose that K = F (x) is a finite extension of even degree over F. Then Ker(r_{K/F}) ⊂ ann_{W (F)}(⟨⟨N_{K/F} (x)⟩⟩).

Proof. Let s be the F -linear functional in (20.8). By Corollary 20.6 and Lemma 20.9, we have

Ker(r_{K/F} : W (F) → W (K)) ⊂ ann_{W (F)}(s_{∗}(⟨1⟩)) = ann_{W (F)}(⟨⟨N_{K/F} (x)⟩⟩). □

Corollary 20.11. Let K/F be a finite field extension of odd degree. Then the map r_{K/F} : W (F) → W (K) is injective.

Proof. If K = F (x) and s is as in (20.8), then by Corollary 20.6 and Lemma 20.9, we have

Ker(r_{K/F} : W (F) → W (K)) ⊂ ann_{W (F)}(s_{∗}(⟨1⟩)) = ann_{W (F)}(⟨1⟩) = 0.

The general case follows by induction of the odd integer [K : F]. □

Note that this corollary provides a more elementary proof of Corollary 18.6.

Lemma 20.12. The transfer induced by the F -linear functional s in (20.8) satisfies

s_{∗}(⟨x⟩_{b}) = \{ \begin{matrix}⟨a⟩_{b} & if n is odd, \\ 0 & if n is even. \end{matrix}

Proof. Let 𝔟 = s_{∗}(⟨x⟩). First suppose that n = 2m + 1 is odd. Then

𝔟(x^{i}, x^{j}) = \{ \begin{matrix}0 & if i + j < n − 1, \\ a & if i + j = n − 1. \end{matrix}

It follows that det 𝔟 = (−1)^{m}aF^{×2} and the subspace W ⊂ K spanned by all x^{i} with i ≠ m and 1 ≤ i ≤ n is nondegenerate. In particular, K = W ⊕ W^{⊥} by Proposition 1.6 and W^{⊥} is 1-dimensional by dimension count. Computing determinants, we see that 𝔟|_{W^{⊥}} ≃ ⟨a⟩. As the subspace of W spanned by x^{i}, i ∈ [0, m − 1], is a lagrangian of 𝔟|_{W}, the form 𝔟|_{W} is metabolic. Consequently, 𝔟 = 𝔟|_{W^{⊥}} = ⟨a⟩ in W (F).

Next, suppose that n = 2m is even. The subspace of K spanned by x^{i}, i ∈ [0, m − 1], is a lagrangian of 𝔟 so 𝔟 is metabolic and 𝔟 = 0 in W (F). □

Corollary 20.13. Let s_{∗} be the transfer induced by the F -linear functional s in (20.8). Then s_{∗}(⟨⟨x⟩⟩) = ⟨⟨a⟩⟩ in W (F).

20.B. Similarity theorems. As a consequence, we get the norm principle first established by Scharlau in [119].

Theorem 20.14 (Similarity Norm Principle). Let K/F be a finite field extension and φ a nondegenerate even-dimensional quadratic form over F. Then

N_{K/F} (G(φ_{K})) ⊂ G(φ).

Proof. Let x ∈ G(φ_{K}). Suppose first that K = F (x). Let s be as in (20.8). As ⟨⟨x⟩⟩ · φ_{K} = 0 in I_{q}(K), applying the transfer s_{∗} : I_{q}(K) → I_{q}(F) yields

0 = s_{∗}(⟨⟨x⟩⟩ · φ_{K}) = s_{∗}(⟨⟨x⟩⟩) · φ = ⟨⟨N_{K/F} (x)⟩⟩ · φ

in I_{q}(F) by Frobenius Reciprocity 20.2 and Corollary 20.13. Hence N_{K/F} (x) ∈ G(φ) by Remark 8.17.

In the general case, set k = [K : F (x)]. If k is even, we have

N_{K/F} (x) = N_{F (x)/F} (x)^{k} ∈ G(φ)

since F^{×2} ⊂ G(φ). If k is odd, the homomorphism I_{q}(F (x)) → I_{q}(K) is injective by Remark 18.6, hence ⟨⟨x⟩⟩·φ_{F (x)} = 0. By the first part of the proof, N_{F (x)/F} (x) ∈ G(φ). Therefore, N_{K/F} (x) ∈ N_{F (x)/F} (x)F^{×2} ⊂ G(φ). □

We turn to similarities of forms over polynomial rings. As with values, Knebusch proved analogous results for similarities in [82].

Lemma 20.15. Let φ be a nondegenerate quadratic form of even dimension and p ∈ F [t] a monic irreducible polynomial (in one variable). If φ_{F (p)} is hyperbolic, then p ∈ G(φ_{F (t)}).

Proof. Let x be the image of t in K = F (p) = F [t]/(p). We have p is the norm of t − x in the extension K(t)/F (t). Since φ_{K(t)} is hyperbolic, t − x ∈ G(φ_{K(t)}). Applying the Norm Principle 20.14 to the form φ_{F (t)} and the field extension K(t)/F (t) yields p ∈ G(φ_{F (t)}). □

Theorem 20.16 (Quadratic Similarity Theorem). Let φ be a nondegenerate quadratic form of even dimension and f ∈ F [T] = F [t_{1}, …, t_{n}] a nonzero polynomial. Then the following conditions are equivalent:

(1) f^{∗}f ∈ G(φ_{F (T)}).

(2) There exists an a ∈ F^{×} such that af ∈ G(φ_{F (T)}).

(3) For any irreducible divisor p of f to an odd power, the form φ_{F (p)} is hyperbolic.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3) follows from Proposition 19.8.

(3) ⇒ (1): We proceed by induction on the number n of variables. We may assume that f is irreducible and deg_{t_{1}} f > 0. In particular, f is an irreducible polynomial in t_{1} over the field E = F (T′) = F (t_{2}, …, t_{n}). Let g ∈ F [T′] be the leading term of f. In particular, g^{∗} = f^{∗}. As the polynomial f′ = fg^{−1} in E[t_{1}] is monic irreducible and E(f′) = F (f), the form φ_{E(f′)} is hyperbolic. Applying Lemma 20.15 to φ_{E} and the polynomial f′, we have fg = f′ · g^{2} ∈ G(φ_{F (T)}).

Let p ∈ F [T′] be an irreducible divisor of g to an odd power. Since p does not divide f, by the first part of the proof applied to the polynomial fg, the form φ_{F (p)(t_{1})} is hyperbolic. Since the homomorphism I_{q}(F (p)) → I_{q}(F (p)(t_{1})) is injective by Remark 8.18, we have φ_{F (p)} is hyperbolic. Applying the induction hypothesis to g yields g^{∗}g ∈ G(φ_{F (T′)}). Therefore, f^{∗}f = g^{∗}f = g^{∗}g · fg · g^{−2}· ∈ G(φ_{F (T)}). □

Theorem 20.17 (Bilinear Similarity Norm Principle). Let K/F be a finite field extension and 𝔟 an anisotropic symmetric bilinear form over F of positive dimension. Then

N_{K/F} (G((𝔟_{K})_{an})) ⊂ G(𝔟).

Proof. Let x ∈ G((𝔟_{K})_{an}). Suppose first that K = F (x). Let s be as in (20.8). Let 𝔟_{K} = (𝔟_{K})_{an} ⊥ 𝔠 with 𝔠 a metabolic form over K. Then x𝔠 is metabolic, so

𝔟_{K} = (𝔟_{K})_{an} = x(𝔟_{K})_{an} = x((𝔟_{K})_{an} + 𝔠) = x𝔟_{K}

in W (K). Consequently, ⟨⟨x⟩⟩·𝔟_{K} = 0 in I(K). Applying the transfer s_{∗} : W (K) → W (F) yields

0 = s_{∗}(⟨⟨x⟩⟩ · 𝔟_{K}) = s_{∗}(⟨⟨x⟩⟩) · 𝔟 = ⟨⟨N_{K/F} (x)⟩⟩ · 𝔟

by Frobenius Reciprocity 20.2 and Corollary 20.13. Hence N_{K/F} (x)𝔟 = 𝔟 in W (F) with both sides anisotropic. It follows from Proposition 2.4 that N_{K/F} (x) ∈ G(𝔟).

In the general case, set k = [K : F (x)]. If k is even, we have

N_{K/F} (x) = N_{F (x)/F} (x)^{k} ∈ G(𝔟)

since F^{×2} ⊂ G(𝔟). If k is odd, the homomorphism W (F (x)) → W (K) is injective by Corollary 18.6, hence ⟨⟨x⟩⟩ · (𝔟_{F (x)})_{an} = 0 in W (F (x)). Thus x ∈ G((𝔟_{F (x)})_{an}) by Proposition 2.4. By the first part of the proof, N_{F (x)/F} (x) ∈ G(𝔟). Consequently, N_{K/F} (x) ∈ N_{F (x)/F} (x)F^{×2} ⊂ G(𝔟). □

Lemma 20.18. Let 𝔟 be a nondegenerate anisotropic symmetric bilinear form and p ∈ F [t] a monic irreducible polynomial (in one variable). If 𝔟_{F (p)} is metabolic, then p ∈ G(𝔟_{F (t)}).

Proof. Let x be the image of t in K = F (p) = F [t]/(p). We have p is the norm of t − x in the extension K(t)/F (t). Since 𝔟_{K(t)} is metabolic, (𝔟_{K(t)})_{an} = 0. Thus x − t ∈ G((𝔟_{K(t)})_{an}). Applying the Norm Principle 20.17 to the anisotropic form 𝔟_{F (t)} and the field extension K(t)/F (t) yields p ∈ G(𝔟_{F (t)}). □

Theorem 20.19 (Bilinear Similarity Theorem). Let 𝔟 be an anisotropic bilinear form of even dimension and f ∈ F [T] = F [t_{1}, …, t_{n}] a nonzero polynomial. Then the following conditions are equivalent:

(1) f^{∗}f ∈ G(𝔟_{F (T)}).

(2) There exists an a ∈ F^{×} such that af ∈ G(𝔟_{F (T)}).

(3) For any irreducible divisor p of f to an odd power, the form 𝔟_{F (p)} is metabolic.

Proof. Let φ = φ_{𝔟} be of dimension m.

(1) ⇒ (2) is trivial.

(2) ⇒ (3): Let p be an irreducible factor of f to an odd degree. As F (T) is the quotient field of the localization F [T]_{(p)} and F [T]_{(p)} is a DVR, we have a group homomorphism ∂ : W (F (T)) → W (F (p)) given by Lemma 19.10. Since p is a divisor to an odd power of f,

𝔟_{F (p)} = ∂(𝔟_{F (T)}) = ∂(af𝔟_{F (T)}) = 0

in W (F (p)). Thus 𝔟_{F (p)} is metabolic.

(3) ⇒ (1): The proof is analogous to the proof of (3) ⇒ (1) in the Quadratic Similarity Theorem 20.16 with Lemma 20.18 replacing Lemma 20.15 and hyperbolicity replaced by metabolicity. □

Corollary 20.20. Let φ be a quadratic form (respectively, 𝔟 an anisotropic bilinear form) on V over F and f ∈ F [T] with T = (t_{1}, …, t_{n}). Suppose that f ∈ G(φ_{F (T)}) (respectively, f ∈ G(𝔟_{F (T)})). If f(a) is defined and nonzero with a ∈ F^{n}, then f(a) ∈ G(φ).

Proof. We may assume that φ is anisotropic as G(φ) = G(φ_{an}). (Cf. Remark 8.9.) By induction, we may assume that f is a polynomial in one variable t. Let R = F [t]_{(t−a)}, a DVR. As f(a) ≠ 0, we have f ∈ R^{×}. Over F (t), we have φ_{F (t)} ≃ fφ_{F (t)}, hence φ_{R} ≃ fφ_{R} by Proposition 19.9. Since F is the residue class field of R, upon taking the residue forms, we see that φ = f(a)φ as needed.

As in the quadratic case, we reduce to f being a polynomial in one variable. We then have 𝔟_{F (t)} ≃ f𝔟_{F (t)}. Taking ∂ of this equation relative to the DVR R = F [t]_{(t−a)} yields 𝔟̄ = f̄𝔟̄ = f(a)𝔟̄ in W (F) as f ∈ R^{×}. The result follows by Proposition 2.4. □

Corollary 20.21. Let φ be an quadratic form (respectively, 𝔟 an anisotropic bilinear form) on V over F and g ∈ F [T]. Suppose that g ∈ G(φ_{F (T)}) (respectively, g ∈ G(𝔟_{F (T)})). Then g^{∗} ∈ G(φ) (respectively, g^{∗} ∈ G(𝔟)).

Proof. We may assume that φ is anisotropic as G(φ) = G(φ_{an}). (Cf. Remark 8.9.) By induction on the number of variables, we may assume that g ∈ F [t]. By Lemma 18.1 and Lemma 9.1, we must have deg g = 2r is even. Let h(t) = t^{2r}g(1/t) ∈ G(φ_{F (t)}). Then g^{∗} = h(0) ∈ G(φ) by Corollary 20.20. An analogous proof shows the result for symmetric bilinear forms (using also Lemma 9.2 to see that deg g is even). □

21. An exact sequence for W (F (t))

Let 𝔸_{F}^{1} be the 1-dimensional affine line over F. Let x ∈ 𝔸_{F}^{1} be a closed point and F (x) be the residue field of x. Then there exists a unique monic irreducible polynomial f_{x} ∈ F [t] of degree d = deg x such that F (x) = F [t]/(f_{x}). By Lemma 19.10, we have the first and second residue homomorphisms with respect to the DVR O_{𝔸_{F}^{1} ,x} and prime element f_{x}:

W (F (t)) ⟶^{∂} W (F (x)) and W (F (t)) ⟶^{∂_{fx}} W (F (x)).

Denote ∂_{f_{x}} by ∂_{x}. If g ∈ F [t], then ∂_{x}(⟨g⟩) = 0 unless f_{x} | g in F [t]. It follows if 𝔟 is a nondegenerate bilinear form over F (t) that ∂_{x}(𝔟) = 0 for almost all x ∈ 𝔸_{F}^{1} .

We have:

21. AN EXACT SEQUENCE FOR W (F (t)) 89

Theorem 21.1. The sequence

0 → W (F) ⟶^{r_{F (t)/F}} W (F (t)) ⟶^{∂} ∐_{x∈𝔸_{F}^{1}} W (F (x)) → 0

is split exact where ∂ = (∂_{x}).

Proof. As anisotropic bilinear forms remain anisotropic under a purely transcendental extension, r_{F (t)/F} is monic. It is split by the first residue homomorphism with respect to any rational point in 𝔸_{F}^{1} .

Let F [t]_{d} := {g | g ∈ F [t], deg g ≤ d} and L_{d} ⊂ W (F (t)) the subring generated by ⟨g⟩ with g ∈ F [t]_{d}. Then L_{0} ⊂ L_{1} ⊂ L_{2} ⊂ ⋯ and W (F (t)) = ⋃_{d} L_{d}. Note that im(r_{F (t)/F}) = L_{0}. Let S_{d} be the multiplicative monoid in F [t] generated by F [t]_{d} \ {0}. As a group L_{d} is generated by 1-dimensional forms of the type

(21.2)  ⟨f_{1} ⋯ f_{m}g⟩

with distinct monic irreducible polynomials f_{1}, …, f_{m} ∈ F [t] of degree d and g ∈ S_{d−1}.

Claim: The additive group L_{d}/L_{d−1} is generated by ⟨fg⟩ + L_{d−1} with f ∈ F [t] monic irreducible of degree d and g ∈ S_{d−1}. Moreover, if h ∈ F [t]_{d−1} satisfies g ≡ h mod (f), then ⟨fg⟩ ≃ ⟨fh⟩ mod L_{d−1}.

We first must show that a generator of the form in (21.2) is a sum of the desired forms mod L_{d−1}. By induction on m, we need only do the case m = 2. Let f_{1}, f_{2} be distinct irreducible monic polynomials of degree d and g ∈ S_{d−1}. Let h = f_{1}−f_{2}, so deg h < d. We have

⟨f_{1}⟩ = ⟨h⟩ + ⟨f_{2}⟩ − ⟨f_{1}f_{2}h⟩

in W (F (t)) by the Witt relation (4.2). Multiplying this equation by ⟨f_{2}g⟩ and deleting squares, yields

⟨f_{1}f_{2}g⟩ = ⟨f_{2}gh⟩ + ⟨g⟩ − ⟨f_{1}gh⟩ ≡ ⟨f_{2}gh⟩ − ⟨f_{1}gh⟩ mod L_{d−1}

as needed.

Now suppose that g = g_{1}g_{2} with g_{1}, g_{2} ∈ F [t]_{d−1}. As f |̸ g by the Division Algorithm, there exist polynomials q, h ∈ F [t] with h ≠ 0 and deg h < d satisfying g = fq + h. It follows that deg q < d. By the Witt relation (4.2), we have

⟨g⟩ = ⟨fq⟩ + ⟨h⟩ − ⟨fqhg⟩

in L_{d}, hence multiplying by ⟨f⟩, we have

⟨fg⟩ = ⟨q⟩ + ⟨fh⟩ − ⟨qhg⟩ ≡ ⟨fh⟩ mod L_{d−1}.

The Claim now follows by induction on the number of factors for a general g ∈ S_{d−1}. Let x ∈ 𝔸_{F}^{1} be of degree d and f = f_{x}. Define

α_{x} : W (F (x)) → L_{d}/L_{d−1} by ⟨g + (f)⟩ ↦ ⟨g⟩ + L_{d−1} for g ∈ F [t]_{d−1}.

We show this map is well-defined. If h ∈ F [t]_{d−1} satisfies gh^{2} ≡ l mod (f), with l ∈ F [t]_{d−1}, then ⟨fg⟩ = ⟨fgh^{2}⟩ ≡ ⟨fl⟩ mod L_{d−1} by the Claim, so the map is well-defined on 1-dimensional forms. If g_{1}, g_{2} ∈ F [t]_{d−1} satisfy g_{1} + g_{2} ≠ 0 and h ≡ (g_{1} + g_{2})g_{1}g_{2} mod (f), then

⟨fg_{1}⟩ + ⟨fg_{2}⟩ = ⟨f(g_{1} + g_{2})⟩ + ⟨fg_{1}g_{2}(g_{1} + g_{2})⟩ ≡ ⟨f(g_{1} + g_{2})⟩ + ⟨fh⟩ mod L_{d−1}

by the Claim. As ⟨f⟩ + ⟨−f⟩ = 0 in W (F (t)), it follows that α_{x} is well-defined by Theorem 4.8.

Let x′ ∈ 𝔸_{F}^{1} with deg x′ = d. Then the composition

W (F (x)) ⟶^{α_{x}} L_{d}/L_{d−1} ⟶^{∂_{x′}} W (F (x′))

is the identity if x = x′, otherwise it is the zero map. It follows that the map

∐_{deg x = d}W (F (x)) ⟶^{(α_{x})} L_{d}/L_{d−1}

is split by (∂_{x})_{deg x=d}. As (α_{x}) is surjective by the Claim, it is an isomorphism with inverse (∂_{x})_{deg x=d}. By induction on d, we check that

(∂_{x})_{deg x≤d} : L_{d}/L_{0} ⟶ ∐_{deg x≤d} W (F (x))

is an isomorphism. As L_{0} = W (F), passing to the limit yields the result. □

It follows from Lemma 19.14 or Example 19.13 that ∂_{x}(I^{n}(F (t))) ⊂ I^{n−1}(F (x)) for every x ∈ 𝔸_{F}^{1} .

Corollary 21.3. The sequence

0 → I^{n}(F) ⟶^{r_{F (t)/F}} I^{n}(F (t)) ⟶^{∂} ∐_{x∈𝔸_{F}^{1}} I^{n−1}(F (x)) → 0

is split exact for each n ≥ 1.

Proof. We show by induction on d = deg x that I^{n−1}(F (x)) lies in im(∂). Let g_{2}, …, g_{n} ∈ F [t] be of degree < d. We need to prove that 𝔟 = ⟨⟨ḡ_{2}, …, ḡ_{n}⟩⟩ lies in im(∂) where ḡ_{i} is the image of g_{i} in F (x). By Example 19.13, we have ∂_{x}(𝔠) = 𝔟 where 𝔠 = ⟨⟨−f_{x}, g_{2}, …, g_{n}⟩⟩. Moreover, 𝔠 − 𝔟 ∈ ∐_{deg x<d} I^{n−1}(F (x)) and therefore 𝔠 − 𝔟 ∈ im(∂) by induction.

To finish, it suffices to show exactness at I^{n}(F (t)). Let 𝔟 ∈ Ker(∂). By Theorem 21.1, there exists 𝔠 ∈ W (F) satisfying r_{F (t)/F} (𝔠) = 𝔟. We show 𝔠 ∈ I^{n}(F). Let x ∈ 𝔸_{F}^{1} be a fixed rational point and f = t−t(x). Define ρ : W (F (t)) → W (F) by ρ(𝔡) = ∂_{x}(⟨⟨−f⟩⟩ · 𝔡). By Lemma 19.14, we have ρ(I^{n}(F (t)) ⊂ I^{n}(F) as F (x) = F. By Example 19.13, the composition ρ ∘ r_{F (t)/F} is the identity. It follows that 𝔠 = ρ(𝔟) ∈ I^{n}(F) as needed. □

The above was proven by Milnor in [106] for fields of characteristic not 2. We wish to modify the sequence in Theorem 21.1 to the projective line ℙ_{F}^{1} . If x ∈ 𝔸_{F}^{1} is of degree n, let s_{x} : F (x) → F be the F -linear functional

s_{x}(t^{n−1}(x)) = 1 and s_{x}(t^{i}(x)) = 0 for i < n − 1.

The infinite point ∞ corresponds to the 1/t-adic valuation. It has residue field F. The corresponding second residue homomorphism ∂_{∞} : W (F (t)) → W (F) is taken with respect to the prime 1/t. So if 0 ≠ h ∈ F [t] is of degree n and has leading coefficient a, we have ∂_{∞}(⟨h⟩) = ⟨a⟩ if n is odd, and ∂_{∞}(⟨h⟩) = 0 otherwise. Define (s_{∞})_{∗} to be −Id : W (F) → W (F). The following theorem was first proved by Scharlau in [120] for fields of characteristic not 2.

Theorem 21.4. The sequence

0 → W (F) ⟶^{r_{F (t)/F}} W (F (t)) ⟶^{∂} ∐_{x∈ℙ_{F}^{1}} W (F (x)) ⟶^{s_{∗}} W (F) → 0

is exact with ∂ = (∂_{x}) and s_{∗} = ((s_{x})_{∗}).

Proof. The map (s_{∞})_{∗} is −Id. Hence by Theorem 21.1, it suffices to show s_{∗} ∘ ∂ is the zero map.

As 1-dimensional bilinear forms generate W (F (t)), it suffices to check the result on 1-dimensional forms. Let ⟨af_{1} ⋯ f_{n}⟩ be a 1-dimensional form with f_{i} ∈ F [t] monic of degree d_{i} and a ∈ F^{×} for i ∈ [1, n]. Let x_{i} ∈ 𝔸_{F}^{1} satisfy f_{i} = f_{x_{i}} and s_{i} = s_{x_{i}} for 1 ≤ i ≤ n. We must show that

∑_{X∈𝔸_{F}^{1}} (s_{x})_{∗} ∘ ∂_{x}(⟨af_{1} ⋯ f_{n}⟩) = −(s_{∞})_{∗} ∘ ∂_{∞}(⟨af_{1} ⋯ f_{n}⟩)

in W (F). Multiplying through by ⟨a⟩, we may also assume that a = 1.

Set A = F [t]/(f_{1} ⋯ f_{n}) and d = dim A. Then d = ∑ d_{i}. Let : F [t] → A be the canonical epimorphism and set q_{i} = (f_{1} ⋯ f_{n})/f_{i}. We have an F -vector space homomorphism

α : ∐_{i=1}^{n} F (x_{i}) → A given by (h_{1}(x_{i}), …, h_{n}(x_{i})) ↦ ∑ h̄_{i}q̄_{i} for all h ∈ F [t].

We show that α is an isomorphism. As both spaces have the same dimension, it suffices to show α is monic. As the q_{i} are relatively prime in F [t], we have an equation ∑_{i=1}^{n} g_{i}q_{i} = 1 with all g_{i} ∈ F [t]. Then the map

A → ∐ F (x_{i}) given by h̄ → (h(x_{1})g(x_{1}), …, h(x_{n})g_{n}(x_{n}))

splits α, hence α is monic as needed. Set A_{i} = α(F (x_{i})) for 1 ≤ i ≤ n.

Let s : A → F be the F -linear functional defined by s(t̄^{d−1}) = 1 and s(t̄^{i}) = 0 for 0 ≤ i < d − 1. Define 𝔟 to be the bilinear form on A over F given by 𝔟(f̄, h̄) = s(f̄h̄) for f, h ∈ F [t]. If i ≠ j, we have

𝔟(α(f(x_{i})), α(h(x_{j}))) = 𝔟(f̄q̄_{i}, h̄q̄_{j}) = s(f̄h̄q̄_{i}q̄_{j}) = s(0) = 0

for all f, h ∈ F [t]. Consequently, 𝔟|_{A_{i}} is orthogonal to 𝔟|_{A_{j}} if i ≠ j.

Claim: 𝔟|_{A_{i}} ≃ (s_{i})_{∗}(∂_{f_{i}}(⟨f_{1} ⋯ f_{n}⟩)) for i ∈ [1, n].

Let g, h ∈ F [t]. Write

q_{i}gh = c_{0} + ⋯ + c_{d_{i}−1}t^{d_{i}−1} + f_{i}p

for some c_{i} ∈ F and p ∈ F [t].

By definition, we have

(s_{i})_{∗}(∂_{f_{i}}(⟨f_{1} ⋯ f_{n}⟩))(g(x_{i}), h(x_{i})) = s_{i}(q_{i}(x_{i})g(x_{i})h(x_{i})) = c_{d_{i}−1}.

As deg q_{i} = d − d_{i}, we have deg q_{i}t^{d_{i}−1} = d − 1. Thus

𝔟|_{A_{i}}(α(g(x_{i})), α(h(x_{i}))) = 𝔟(ḡq̄_{i}, h̄q̄_{i}) = s(q̄_{i}^{2}ḡh̄) = c_{d_{i}−1},

and the claim is established.

As ∂_{f}(f_{1} ⋯ f_{n}) = 0 for all irreducible monic polynomials f ≠ f_{i}, i ∈ [1, n], in F [t], we have, by the Claim,

𝔟 = ∑_{i=1}^{n}(s_{i})_{∗}(∂_{x_{i}}(⟨f_{1} ⋯ f_{n}⟩)) = ∑_{x∈𝔸_{F}^{1}} (s_{x})_{∗}(∂_{x}(⟨f_{1} ⋯ f_{n}⟩))

in W (F).

Suppose that d = 2e is even. The form 𝔟 is then metabolic as it has a totally isotropic subspace of dimension e spanned by 1, t̄, …, t̄^{e−1}. We also have (s_{∞})_{∗} ∘ ∂_{∞}(𝔟) = 0 in this case.

Suppose that d = 2e + 1. Then 𝔟 has a totally isotropic subspace spanned by 1, t̄, …, t̄^{e−1} so 𝔟 ≃ ⟨a⟩ ⊥ 𝔠 with 𝔠 metabolic by the Witt Decomposition Theorem 1.27. Computing det 𝔟 on the basis {1, t̄, …, t̄^{d−1}}, we see that ⟨a⟩ = ⟨1⟩. As (s_{∞})_{∗} ∘ ∂_{∞}(𝔟) = −⟨1⟩, the result follows. □

Corollary 21.5. Let K be a finite simple extension of F and s : K → F a nontrivial F -linear functional. Then s_{∗}(I^{n}(K)) ⊂ I^{n}(F) for all n ≥ 0. Moreover, the induced map I^{n}(K)/I^{n+1}(K) → I^{n}(F)/I^{n+1}(F) is independent of the nontrivial F -linear functional s for all n ≥ 0.

Proof. Let x lie in 𝔸_{F}^{1} with K = F (x). Let 𝔟 ∈ I^{n}(K). By Lemma 21.3, there exists 𝔠 ∈ I^{n+1}(F (t)) such that ∂_{y}(𝔠) = 0 for all y ∈ 𝔸_{F}^{1} unless y = x in which case ∂_{x}(𝔠) = 𝔟. It follows by Theorem 21.4 that

0 = ∑_{y∈ℙ_{F}^{1}} (s_{y})_{∗} ∘ ∂_{y}(𝔠) = (s_{x})_{∗}(𝔟) − ∂_{∞}(𝔠).

By Lemma 19.14, we have ∂_{∞}(𝔠) ∈ I^{n}(F), so (s_{x})_{∗}(𝔟) ∈ I^{n}(F). Suppose that s : K → F is another nontrivial F -linear functional. As in the proof of Corollary 20.7, there exists a c ∈ K^{×} such that (s)_{∗}(𝔠) = (s_{x})_{∗}(c𝔠) for all symmetric bilinear forms 𝔠. In particular, (s)_{∗}(𝔟) = (s_{x})_{∗}(c𝔟) lies in I^{n}(F). As ⟨⟨c⟩⟩ · 𝔟 ∈ I^{n+1}(K), we also have

s_{∗}(𝔟) − (s_{x})_{∗}(𝔟) = (s_{x})_{∗}(⟨⟨c⟩⟩ · 𝔟)

lies in I^{n+1}(F). The result follows. □

The transfer induced by distinct nontrivial F -linear functionals K → F are not, in general, equal on I^{n}(F).

Exercise 21.6. Show that Corollary 21.5 holds for arbitrary finite extensions K/F.

Corollary 21.7. The sequence

0 → I^{n}(F) ⟶^{r_{F (t)/F}} I^{n}(F (t)) ⟶^{∂} ∐_{x∈ℙ_{F}^{1}} I^{n−1}(F (x)) ⟶^{s_{∗}} I^{n−1}(F) → 0

is exact.

CHAPTER IV Function Fields of Quadrics

22. Quadrics

A quadratic form φ over F defines a projective quadric X_{φ} over F. The quadric X_{φ} is smooth if and only if φ is nondegenerate (cf. Proposition 22.1). The quadric X_{φ} encodes information about isotropy properties of φ, namely the form φ is isotropic over a field extension E/F if and only if X_{φ} has a point over E. In the third part of the book, we will use algebraic-geometric methods to study isotropy properties of φ.

If 𝔟 is a symmetric bilinear form, the quadric X_{φ_{𝔟}} reflects isotropy properties of 𝔟 (and of φ_{𝔟} as well). If the characteristic of F is 2, only totally singular quadratic forms arise from symmetric bilinear forms. In particular, quadrics arising from bilinear forms are not smooth. Therefore, algebraic-geometric methods have wider application in the theory of quadratic forms than in the theory of bilinear forms.

In the previous sections, we looked at quadratic forms over field extensions determined by irreducible polynomials. In particular, we were interested in when a quadratic form becomes isotropic over such a field. Viewing a quadratic form as a homogeneous polynomial of degree two, results from these sections apply.

Let φ and ψ be two anisotropic quadratic forms. In this section, we begin our study of when φ becomes isotropic or hyperbolic over the function field F (X_{ψ}) of the integral quadric X_{ψ}. It is natural at this point to introduce the geometric language that we shall use, i.e., to associate to a quadratic form a projective quadric.

Let φ be a quadratic form on V. Viewing φ ∈ S^{2}(V^{∗}), we define the projective quadric associated to φ to be the closed subscheme

X_{φ} = Proj (S^{•}(V^{∗})/(φ))

of the projective space ℙ(V) = Proj S^{•}(V^{∗}) where S^{•}(V^{∗}) is the symmetric algebra of the dual space V^{∗} of V. The scheme X_{φ} is equidimensional of dimension dim V −2 if φ ≠ 0 and dim V ≥ 2. We define the Witt index of X_{φ} by i_{0}(X_{φ}) := i_{0}(φ). By construction, for any field extension L/F, the set of L-points X_{φ}(L) coincides with the set of isotropic lines in V_{L}. Therefore, X_{φ}(L) = ∅ if and only if φ_{L} is anisotropic.

For any field extension K/F we have X_{φ_{K}} = (X_{φ})_{K}.

Let ψ be a subform of φ. The inclusion of vector spaces V_{ψ} ⊂ V gives rise to a surjective graded ring homomorphism

S^{•}(V^{∗})/(φ) → S^{•}(V_{ψ}^{∗})/(ψ)

which in turn leads to a closed embedding X_{ψ} ↪ X_{φ}. We shall always identify X_{ψ} with a closed subscheme of X_{φ}.

Proposition 22.1. Let φ be a nonzero quadratic form of dimension at least 2. Then the quadric X_{φ} is smooth if and only if φ is nondegenerate.

Proof. By Lemma 7.16, we may assume that F is algebraically closed. We claim that ℙ(rad φ) is the singular locus of X_{φ}. Let 0 ≠ u ∈ V be an isotropic vector. Then the isotropic line U = F u ⊂ V can be viewed as a rational point of X_{φ}. As φ(u + εv) = 0 if and only if u is orthogonal to v (where ε^{2} = 0), the tangent space T_{X,U} is the subspace Hom(U, U^{⊥}/U) of the tangent space T_{ℙ(V),U} = Hom(U, V/U) (cf. Example 104.20). In particular, the point U is regular on X if and only if dim T_{X,U} = dim X = dim V − 2 if and only if U^{⊥} ≠ V, i.e., U is not contained in rad φ. Thus X_{φ} is smooth if and only if rad φ = 0, i.e., φ is nondegenerate. □

We say that the quadratic form φ on V is irreducible if φ is irreducible in the ring S^{•}(V^{∗}). If φ is nonzero and not irreducible, then φ = l · l′ for some nonzero linear forms l, l′ ∈ V^{∗}. Then rad φ = Ker(l) ∩ Ker(l′) has codimension at most 2 in V. Therefore, the induced form φ̄ on V/ rad φ is either 1-dimensional or a hyperbolic plane. It follows that a regular quadratic form φ is irreducible if and only if dim φ ≥ 3 or dim φ = 2 and φ is anisotropic.

If φ is irreducible, X_{φ} is an integral scheme. The function field F (X_{φ}) is called the function field of φ and will be denoted by F (φ). By definition, F (φ) is the subfield of degree 0 elements in the quotient field of the domain S^{•}(V^{∗})/(φ). Note that the quotient field of S^{•}(V^{∗})/(φ) is a purely transcendental extension of F (φ) of degree 1. Clearly, φ is isotropic over the quotient field of S^{•}(V^{∗})/(φ) and therefore is isotropic over F (φ).

Example 22.2. Let σ be an anisotropic binary quadratic form. As σ is isotropic over F (σ), it follows from Corollary 12.3 that F (σ) ≃ C_{0}(σ).

If K/F is a field extension such that φ_{K} is still irreducible, we simply write K(φ) for K(φ_{K}).

Example 22.3. Let φ and ψ be irreducible quadratic forms. Then

F (X_{φ} × X_{ψ}) ≃ F (φ)(ψ_{F (φ)}) ≃ F (ψ)(φ_{F (ψ)}).

Let φ and ψ be two irreducible regular quadratic forms. We shall be interested in when φ_{F (ψ)} is hyperbolic or isotropic. A consequence of the Quadratic Similarity Theorem 20.16 is:

Proposition 22.4. Let φ be a nondegenerate quadratic form of even dimension and ψ be an irreducible quadratic form of dimension n over F. Suppose that T = (t_{1}, …, t_{n}) and b ∈ D(ψ). Then φ_{F (ψ)} is hyperbolic if and only if

b · ψ(T)φ_{F (T)} ≃ φ_{F (T)}.

Proof. By the Quadratic Similarity Theorem 20.16, we have φ_{F (ψ)} is hyperbolic if and only if ψ^{∗}ψ(T)φ_{F (T)} ≃ φ_{F (T)}. Let b ∈ D(ψ). Choosing a basis for V with first vector v satisfying ψ(v) = b, we have ψ^{∗} = b. □

Theorem 22.5 (Subform Theorem). Let φ be a nonzero anisotropic quadratic form and ψ be an irreducible anisotropic quadratic form such that the form φ_{F (ψ)} is hyperbolic. Let a ∈ D(φ) and b ∈ D(ψ). Then abψ is isometric to a subform of φ and, therefore, dim ψ ≤ dim φ.

Proof. We view ψ as an irreducible polynomial in F [T]. The form φ is nondegenerate of even dimension by Remark 7.18, so by Corollary 22.4, we have bψ(T) ∈ G(φ_{F (T)}). Since a ∈ D(φ), we have abψ(T) ∈ D(φ_{F (T)}). By the Representation Theorem 17.12, we conclude that abψ is a subform of φ. □

By the proof of the theorem and Corollary 20.20, we have

Corollary 22.6. Let φ be an anisotropic quadratic form and ψ an irreducible anisotropic quadratic form. If φ_{F (ψ)} is hyperbolic, then D(ψ)D(ψ) ⊂ G(φ). In particular, if 1 ∈ D(ψ), then D(ψ) ⊂ G(φ).

Remark 22.7. The natural analogues of the Representation Theorem 17.12 and the Subform Theorem 22.5 are not true for bilinear forms in characteristic 2. Let 𝔟 = ⟨1, b⟩ and 𝔠 = ⟨1, c⟩ be anisotropic symmetric bilinear forms with b and c = x^{2} + by^{2} nonzero and bF^{×2} ≠ cF^{×2} in a field F of characteristic 2. Thus 𝔟 ≄ 𝔠. However, φ_{𝔟} ≃ φ_{𝔠} by Example 7.27. So φ_{𝔠}(t_{1}, t_{2}) ∈ D(φ_{𝔟}_{F (t_{1},t_{2})}) and 𝔠_{F (φ_{𝔟})} is isotropic, hence metabolic, but a𝔠 is not a subform of 𝔟 for any a ≠ 0.

We do have, however, the following:

Corollary 22.8. Let 𝔟 and 𝔠 be anisotropic bilinear forms with dim 𝔠 ≥ 2 and 𝔟 nonzero. Let ψ be the associated quadratic form of 𝔠. If 𝔟_{F (ψ)} is metabolic, then dim 𝔠 ≤ dim 𝔟.

Proof. Let φ = φ_{𝔟}. By the Bilinear Similarity Theorem 20.19 and Lemma 9.2, we have aψ(T) ∈ G(𝔟_{F (T)}) ⊂ G(φ_{F (T)}) for some a ∈ F^{×} where T = (t_{1}, …, t_{dim ψ}). It follows that bψ(T) ∈ D(φ_{F (T)}) for some b ∈ F^{×}. Consequently,

dim 𝔟 = dim φ ≥ dim ψ = dim 𝔠

by the Representation Theorem 17.12. □

We turn to the case in which a quadratic form becomes isotropic over the function field of another form or itself.

Proposition 22.9. Let φ be an irreducible regular quadratic form. Then the field extension F (φ)/F is purely transcendental if and only if φ is isotropic.

Proof. Suppose that the field extension F (φ)/F is purely transcendental. As φ_{F (φ)} is isotropic, φ is isotropic by Lemma 7.15.

Now suppose that φ is isotropic. Then φ = ℍ ⊥ φ′ for some φ′ by Proposition 7.13. Let V = V_{φ}, V′ = V_{φ′} and let h, h′ ∈ V be a hyperbolic pair of ℍ. Let ψ = φ|_{F h′⊕V′} with h′ ∈ (V′)^{⊥}. It suffices to show that X_{φ} \ X_{ψ} is isomorphic to an affine space. Every isotropic line in X_{φ} \ X_{ψ} has the form F (h + ah′ + v′) for unique a ∈ F and v′ ∈ V′ satisfying

0 = φ(h + ah′ + v′) = a + φ(v′),

i.e., a = −φ(v′). Therefore, the morphism X_{φ} \ X_{ψ} → 𝔸(V′) taking F (h + ah′ + v′) to v′ is an isomorphism with the inverse given by v′ ↦ F (h − φ(v′)h′ + v′). □

Remark 22.10. Let char F = 2 and let φ be an irreducible totally singular form. Then the field extension F (φ)/F is not purely transcendental even if φ is isotropic.

Proposition 22.11. Let φ be an anisotropic quadratic form over F and K/F a quadratic field extension. Then φ_{K} is isotropic if and only if there is a binary subform σ of φ such that F (σ) ≃ K.

Proof. Let σ be a binary subform of φ with F (σ) ≃ K. Since σ is isotropic over F (σ) we have φ isotropic over F (σ) ≃ K.

Conversely, suppose that φ_{K}(v) = 0 for some nonzero v ∈ (V_{φ})_{K}. Since K is quadratic over F, there is a 2-dimensional subspace U ⊂ V_{φ} with v ∈ U_{K}.

Therefore, the form σ = φ|_{U} is isotropic over K. As σ is also isotropic over F (σ), it follows from Corollary 12.3 and Example 22.2 that F (σ) ≃ C_{0}(σ) ≃ K. □

Corollary 22.12. Let φ be an anisotropic quadratic form and σ a nondegenerate anisotropic binary quadratic form. Then φ ≃ 𝔟 ⊗ σ ⊥ ψ with 𝔟 a nondegenerate symmetric bilinear form and ψ_{F (σ)} anisotropic.

Proof. Suppose that φ_{F (σ)} is isotropic. By Proposition 22.11, there is a binary subform σ′ of φ with F (σ′) = F (σ). By Corollary 12.2 and Example 22.2, we have σ′ is similar to σ. Consequently, there exists an a ∈ F^{×} such that φ ≃ aσ ⊥ ψ for some quadratic form ψ. The result follows by induction on dim φ. □

Recall that a field extension K/F is called separable if there exists and intermediate field E in K/F with E/F purely transcendental and K/E algebraic and separable. We show that regular quadratic forms remain regular after extending to a separable field extension.

Lemma 22.13. Let φ be a regular quadratic form over F and K/F a separable (possibly infinite) field extension. Then φ_{K} is regular.

Proof. We proceed in several steps.

Case 1: [K : F] = 2.

Let v ∈ (V_{φ})_{K} be an isotropic vector. Then v ∈ U_{K} for a 2-dimensional subspace U ⊂ V_{φ} such that φ|_{U} is similar to the norm form N of K/F (cf. Proposition 12.1). As N is nondegenerate, v ∉ rad 𝔟_{φ_{K}}, therefore, rad φ_{K} = 0.

Case 2: K/F is of odd degree or purely transcendental.

We have φ ≃ φ_{an} ⊥ nℍ. The anisotropic part φ_{an} stays anisotropic over K by Springer’s Theorem 18.5 or Lemma 7.15, respectively; therefore φ_{K} is regular.

Case 3: [K : F] is finite.

We may assume that K/F is Galois by Remark 7.14. Then K/F is a tower of odd degree and quadratic extensions.

Case 4: The general case.

In general, K/F is a tower of a purely transcendental and a finite separable extension. □

We turn to the function field of an irreducible quadratic form.

Lemma 22.14. Let φ be an irreducible quadratic form over F. Then there exists a purely transcendental extension E of F with [F (φ) : E] = 2. Moreover, if φ is not totally singular, the field E can be chosen with F (φ)/E separable. In particular, F (φ)/F is separable.

Proof. Let U ⊂ V_{φ} be an anisotropic line. The rational projection f : X_{φ} −−→ ℙ = ℙ(V/U) taking a line U′ to (U + U′)/U is a double cover, so F (φ)/E is a quadratic field extension where E is the purely transcendental extension F (ℙ) of F.

Let τ be the reflection of φ with respect to a nonzero vector in U. Clearly, f(τU′) = f(U′) for every line U′ in X_{φ}. Therefore, τ induces an automorphism of every fiber of f. In particular, τ induces an automorphism of the generic fiber, hence an automorphism ε of the field F (φ) over E.

If φ is not totally singular, we can choose U not in rad 𝔟_{φ}. Then the isometry τ and the automorphism ε are nontrivial. Consequently, the field extension F (φ)/E is separable. □

22.A. Domination relation. Let φ and ψ be anisotropic quadratic forms of dimension at least 2 over F. We say φ dominates ψ and write φ ≻ ψ if φ_{F (ψ)} is isotropic and write φ ≺≻ ψ if φ ≻ ψ and ψ ≻ φ. For example, if ψ is a subform of φ, then φ ≻ ψ.

We have φ ≻ ψ if and only if there exists a rational map X_{ψ} −−→ X_{φ}.

We show that the relation ≻ is transitive.

Lemma 22.15. Let φ and ψ be anisotropic quadratic forms over F. If ψ ≻ μ, then there exist a purely transcendental field extension E/F and a binary subform σ of ψ_{E} over E such that E(σ) = F (μ).

Proof. By Lemma 22.14, there exists a purely transcendental field extension E/F such that F (μ) is a quadratic extension of E. As ψ is isotropic over F (μ), it follows from Proposition 22.11 applied to the form ψ_{E} and the quadratic extension F (μ)/E that ψ_{E} contains a binary subform σ over E satisfying E(σ) = F (μ). □

Proposition 22.16. Let φ, ψ, and μ be anisotropic quadratic forms over F. If φ ≻ ψ ≻ μ, then φ ≻ μ.

Proof. Consider first the case when μ is a subform of ψ.

We may assume that μ is of codimension one in ψ. Let T = (t_{1}, …, t_{n}) be the coordinates in V_{ψ} so that V_{μ} is given by t_{1} = 0. By assumption, there is v ∈ V_{φ}[T] such that φ(v) is divisible by ψ(T) but v is not divisible by ψ(T). Since ψ is anisotropic, we have deg_{t_{i}} ψ = 2 for every i. Applying the division algorithm by dividing v by ψ with respect to the variable t_{2}, we may assume that deg_{t_{2}} v ≤ 1. Moreover, dividing out a power of t_{1} if necessary, we may assume that v is not divisible by t_{1}. Therefore, the vector w := v|_{t_{1}=0} ∈ V_{φ}[T′] with T′ = (t_{2}, …, t_{n}) is not zero. As deg_{t_{2}} w ≤ 1 and deg_{t_{2}} μ = 2, the vector w is not divisible by μ(T′). On the other hand, φ(w) is divisible by ψ(T)|_{t_{1}=0} = μ(T′), i.e., φ is isotropic over F (μ).

Now consider the general case. By Lemma 22.15, there exist a purely transcendental field extension E/F and a binary subform σ of ψ_{E} over E such that E(σ) = F (μ). By the first part of the proof applied to the forms φ_{E} ≻ ψ_{E} ≻ σ, we have φ_{E} is isotropic over E(σ) = F (μ), i.e., φ ≻ μ. □

Corollary 22.17. Let φ, ψ, and μ be anisotropic quadratic forms over F. If φ ≺≻ ψ, then μ_{F (φ)} is isotropic if and only if μ_{F (ψ)} is isotropic.

Proposition 22.18. Let ψ and μ be anisotropic quadratic forms over F satisfying ψ ≻ μ. Let φ be a quadratic form such that φ_{F (ψ)} is hyperbolic. Then φ_{F (μ)} is hyperbolic.

Proof. Consider first the case when μ is a subform of ψ. Choose variables T′ of μ and variables T = (T′, T′′) of ψ so that μ(T′) = ψ(T′, 0). As φ_{F (ψ)} is hyperbolic, by the Quadratic Similarity Theorem 20.16, we have φ_{F (T)} ≃ aψ(T)φ_{F (T)} over F (T) for some a ∈ F^{×}. Specializing the variables T′′ = 0, we see by Corollary 20.20 that φ_{F (T′)} ≃ aμ(T′)φ_{F (T′)} over F (T′), and again it follows from the Quadratic Similarity Theorem 20.16 that φ_{F (μ)} is hyperbolic.

Now consider the general case. By Lemma 22.15, there exist a purely transcendental field extension E/F and a binary subform σ of ψ_{E} over E such that E(σ) = F (μ). As φ_{E(ψ)} is hyperbolic, by the first part of the proof applied to the forms ψ_{E} ≻ σ, we have φ_{E(σ)} = φ_{F (μ)} is hyperbolic. □

23. Quadratic Pfister forms II

The introduction of function fields of quadrics allows us to determine the main characterization of general quadratic Pfister forms first proven by Pfister in [108] for fields of characteristic different from 2. They are precisely those forms that become hyperbolic over their function fields. In particular, Pfister forms can be characterized as universally round forms.

If φ is an anisotropic general quadratic Pfister form, then φ_{F (φ)} is isotropic, hence hyperbolic by Corollary 9.10. We wish to show the converse of this property. We begin by looking at subforms of Pfister forms.

Lemma 23.1. Let φ be an anisotropic quadratic form over F and ρ a subform of φ. Suppose that D(φ_{K}) and D(ρ_{K}) are groups for all field extensions K/F. Let a = −φ(v) for some v ∈ V_{ρ}^{⊥} \ V_{ρ}. Then the form ⟨⟨a⟩⟩ ⊗ ρ is isometric to a subform of φ.

Proof. Let T = (t_{1}, …, t_{n}) and T′ = (t_{n+1}, …, t_{2n}) be 2n independent variables where n = dim ρ. We have

ρ(T) − aρ(T′) = ρ(T′)[\frac{ρ(T)}{ρ(T′)} − a].

As D(ρ_{F (T,T′)}) is a group, we have \frac{ρ(T)}{ρ(T′)} ∈ D(ρ_{F (T,T′)}), hence \frac{ρ(T)}{ρ(T′)} − a ∈

D(φ_{F (T,T′)}). As ρ(T′) ∈ D(φ_{F (T,T′)}), we have

ρ(T) − aρ(T′) ∈ D(φ_{F (T,T′)})D(φ_{F (T,T′)}) = D(φ_{F (T,T′)}).

By the Representation Theorem 17.12, we conclude that ⟨⟨a⟩⟩ ⊗ ρ is a subform of φ. □

Theorem 23.2. Let φ be a nondegenerate (respectively, totally singular) anisotropic quadratic form over F of dimension n ≥ 1. Let T = (t_{1}, …, t_{n}) and T′ = (t_{n+1}, …, t_{2n}) be 2n independent variables. Then the following are equivalent:

(1) n = 2^{k} for some k ≥ 1 and φ ∈ P_{k}(F) (respectively, φ is a quadratic quasi-Pfister form).

(2) G(φ_{K}) = D(φ_{K}) for all field extensions K/F.

(3) D(φ_{K}) is a group for all field extensions K/F.

(4) Over the rational function field F (T, T′), we have

φ(T)φ(T′) ∈ D(φ_{F (T,T′)}).

(5) φ(T) ∈ G(φ_{F (T)}).

Proof. (2) ⇒ (3) ⇒ (4) are trivial.

(5) ⇐ (1) ⇒ (2): As quadratic Pfister forms are round by Corollary 9.9 and quasi-Pfister forms are round by Corollary 10.3, the implications follow.

(5) ⇒ (4): We have φ(T) ∈ G(φ_{F (T)}) ⊂ G(φ_{F (T,T′)}) and φ(T′) ∈ D(φ_{T,T′)}). It follows by Lemma 9.1 that φ(T)φ(T′) ∈ D(φ_{F (T,T′)}).

(4) ⇒ (3): If K/F is a field extension, then φ(T)φ(T′) ∈ D(φ_{K(T,T′)}). By the Substitution Principle 17.7, it follows that D(φ_{K}) is a group.

(3) ⇒ (1): As 1 ∈ D(φ), it is sufficient to show that φ is a general quadratic (quasi-) Pfister form. We may assume that dim φ ≥ 2. If φ is nondegenerate, φ contains a nondegenerate binary subform, i.e., a 1-fold general quadratic Pfister form. Let ρ be the largest quadratic general Pfister subform of φ if φ is nondegenerate and the largest quasi-Pfister form if φ is totally singular. Suppose that ρ ≠ φ. If φ is nondegenerate, then V_{ρ}^{⊥} ≠ 0 and V_{ρ}^{⊥} ∩ V_{ρ} = rad 𝔟_{ρ} = 0 and if φ is totally singular, then V_{ρ}^{⊥} = V_{φ} and V_{ρ} ≠ V_{φ}. In either case, there exists a v ∈ V_{ρ}^{⊥} \ V_{ρ}. Set a = −φ(v). By Lemma 23.1, we have ⟨⟨a⟩⟩ ⊗ ρ is isometric to a subform of φ, a contradiction. □

Remark 23.3. Let φ be a nondegenerate isotropic quadratic form over F. As hyperbolic quadratic forms are universal and round, if φ is hyperbolic, then φ(T) ∈ G(φ_{F (T)}). Conversely, suppose φ(T) ∈ G(φ_{F (T)}). As

(φ_{F (T)})_{an} ⊥ 𝔦_{0}(φ)ℍ ≃ φ_{F (T)} ≃ φ(T)φ_{F (T)} ≃ φ(T)(φ_{F (T)})_{an} ⊥ 𝔦_{0}(φ)φ(T)ℍ,

we have φ(T) ∈ G((φ_{F (T)})_{an}) by Witt Cancellation 8.4. If φ was not hyperbolic, then the Subform Theorem 22.5 would imply dim φ_{F (T)} ≤ dim(φ_{F (T)})_{an}, a contradiction. Consequently, φ(T) ∈ G(φ_{F (T)}) if and only if φ is hyperbolic.

Corollary 23.4. Let φ be a nondegenerate anisotropic quadratic form of dimension at least two over F. Then the following are equivalent:

(1) dim φ is even and 𝔦_{1}(φ) = dim φ/2.

(2) φ_{F (φ)} is hyperbolic.

(3) φ ∈ GP_{n}(F) for some n ≥ 1.

Proof. Statements (1) and (2) are both equivalent to φ_{F (φ)} contains a totally isotropic subspace of dimension \frac{1}{2} dim φ. Let a ∈ D(φ). Replacing φ by ⟨a⟩φ we may assume that φ represents one. By Theorem 22.4, condition (2) in the corollary is equivalent to condition (5) of Theorem 23.2, hence conditions (2) and (3) above are equivalent. □

Corollary 23.5. Let φ and ψ be quadratic forms over F with φ ∈ P_{n}(F) anisotropic. Suppose that there exists an F -isomorphism F (φ) ≃ F (ψ). Then there exists an a ∈ F^{×} such that ψ ≃ aφ over F, i.e., φ and ψ are similar over F.

Proof. As φ_{F (φ)} is hyperbolic, so is φ_{F (ψ)}. In particular, aψ is a subform of φ for some a ∈ F^{×} by the Subform Theorem 22.5. Since F (φ) ≃ F (ψ), we have dim φ = dim ψ and the result follows. □

In general, the corollary does not generalize to non-Pfister forms. Let F = ℚ(t_{1}, t_{2}, t_{3}). The quadratic forms φ = ⟨⟨t_{1}, t_{2}⟩⟩ ⊥ ⟨−t_{3}⟩ and φ = ⟨⟨t_{1}, t_{3}⟩⟩ ⊥ ⟨−t_{2}⟩ have isomorphic function fields but are not similar. (Cf. [89, Th. XII.2.15].)

Let r : F → K be a homomorphism of fields. Denote the kernel of r_{K/F} : W (F) → W (K) by W (K/F) and the kernel of r_{K/F} : I_{q}(F) → I_{q}(K) by I_{q}(K/F). If φ is a nondegenerate even-dimensional quadratic form over F, we denote by W (F)φ the cyclic W (F)-module in I_{q}(F) generated by φ.

Corollary 23.6. Let φ be an anisotropic quadratic n-fold Pfister form with n ≥ 1 and ψ an anisotropic quadratic form of even dimension over F. Then there is an isometry ψ ≃ 𝔟 ⊗ φ over F for some symmetric bilinear form 𝔟 over F if and only if ψ_{F (φ)} is hyperbolic. In particular, I_{q}(F (φ)/F) = W (F)φ.

Proof. If 𝔟 is a bilinear form, then (𝔟 ⊗ φ)_{F (φ)} = 𝔟_{F (φ)} ⊗ φ_{F (φ)} is hyperbolic by Lemma 8.16 as φ_{F (φ)} is hyperbolic by Corollary 9.10. Conversely, suppose that ψ_{F (φ)} is hyperbolic. We induct on dim ψ. Assume that dim ψ > 0. By the Subform Theorem 22.5 and Proposition 7.22, we have ψ ≃ aφ ⊥ γ for some a ∈ F^{×} and quadratic form γ. The form γ also satisfies γ_{F (φ)} is hyperbolic, so the result follows by induction. □

23.A. The Hauptsatz. We next prove a fundamental fact about forms in I^{n}(F) and I_{q}^{n}(F) due to Arason and Pfister known as the Hauptsatz and proven in [10].

Theorem 23.7 (Hauptsatz). (1) Let 0 ≠ φ be an anisotropic quadratic form lying in I_{q}^{n}(F). Then dim φ ≥ 2^{n}.

(2) Let 0 ≠ 𝔟 be an anisotropic bilinear form lying in I^{n}(F). Then dim(𝔟) ≥ 2^{n}.

Proof. (1): As I_{q}^{n}(F) is additively generated by general quadratic n-fold Pfister forms, we can write φ = ∑_{1=i}^{r} a_{i}ρ_{i} in W (F) for some anisotropic ρ_{i} ∈ P_{n}(F) and a_{i} ∈ F^{×}. We prove the result by induction on r. If r = 1, the result is trivial as ρ_{1} is anisotropic, so we may assume that r > 1. As (ρ_{r})_{F (ρ_{r})} is hyperbolic by Corollary 9.10, applying the restriction map r_{F (ρ_{r})/F} : W (F) → W (F (ρ_{r})) to φ yields φ_{F (ρ_{r})} = ∑_{i=1}^{r−1} a_{i}(ρ_{i})_{F (ρ_{r})} in I_{q}^{n}(F (ρ)). If φ_{F (ρ_{r})} is hyperbolic, then 2^{n} = dim ρ ≤ dim φ by the Subform Theorem 22.5. If this does not occur, then by induction 2^{n} ≤ dim(φ_{F (ρ_{r})})_{an} ≤ dim φ and the result follows.

(2): As I^{n}(F) is additively generated by bilinear n-fold Pfister forms, we can write 𝔟 = ∑_{1=i}^{r} ε_{i}𝔠_{i} in W (F) for some 𝔠_{i} anisotropic bilinear n-fold Pfister forms and ε_{i} ∈ {±1}. Let φ = φ_{𝔠_{r}} be the quadratic form associated to 𝔠_{r}. Then φ_{F (φ)} is isotropic, hence (𝔠_{r})_{F (φ)} is isotropic, hence metabolic by Corollary 6.3. If 𝔟_{F (φ)} is not metabolic, then 2^{n} ≤ dim(𝔟_{F (φ)})_{an} ≤ dim 𝔟 by induction on r. If 𝔟_{F (φ)} is metabolic, then 2^{n} = dim 𝔠 ≤ dim 𝔟 by Corollary 22.8. □

An immediate consequence of the Hauptsatz is a solution to a problem of Milnor, viz.,

Corollary 23.8. ⋂_{i=1}^{∞} I^{n}(F) = 0 and ⋂_{i=1}^{∞} I_{q}^{n}(F) = 0.

The proof of the Hauptsatz for bilinear forms completes the proof of Corollary 6.19 and Theorem 6.20. We have an analogous result for quadratic Pfister forms.

Corollary 23.9. Let φ, ψ ∈ GP_{n}(F). If φ ≡ ψ mod I_{q}^{n+1}(F), then φ ≃ aψ for some a ∈ F^{×}, i.e., φ and ψ are similar over F. If, in addition, D(φ) ∩ D(ψ) ≠ ∅, then φ ≃ ψ.

Proof. By the Hauptsatz 23.7, we may assume both φ and ψ are anisotropic. As ⟨⟨a⟩⟩ ⊗ ψ ∈ GP_{n+1}(F), we have aψ ≡ ψ mod I_{q}^{n+1}(F) for any a ∈ F^{×}. Choose a ∈ F^{×} such that φ ⊥ −aψ in I_{q}^{n+1}(F) is isotropic. By the Hauptsatz 23.7, the form φ ⊥ −aψ is hyperbolic, hence φ = aψ in I_{q}(F). As both forms are anisotropic, it follows by dimension count that φ ≃ aψ by Remark 8.17. If D(φ) ∩ D(ψ) ≠ ∅, then we can take a = 1. □

If φ is a nonzero subform of dimension at least two of an anisotropic quadratic form ρ, then ρ_{F (φ)} is isotropic. As φ must also be anisotropic, we have ρ ≻ φ. For general Pfister forms, we can say more. Let ρ be an anisotropic general quadratic Pfister form. Then ρ_{F (ρ)} is hyperbolic, so it contains a totally isotropic subspace of dimension (dim ρ)/2. Suppose that φ is a subform of ρ satisfying dim φ > (dim ρ)/2. Then φ_{F (ρ)} is isotropic, hence φ ≻ ρ also. This motivates the following:

Definition 23.10. An anisotropic quadratic form φ is called a Pfister neighbor if there is a general quadratic Pfister form ρ such that φ is isometric to a subform of ρ and dim φ > (dim ρ)/2.

For example, nondegenerate anisotropic forms of dimension at most 3 are Pfister neighbors.

Remark 23.11. Let φ be a Pfister neighbor isometric to a subform of a general quadratic Pfister form ρ with dim φ > (dim ρ)/2. By the above, φ ≺≻ ρ. Let σ be another general quadratic Pfister form such that φ is isometric to a subform of σ and dim φ > (dim σ)/2. As ρ ≺≻ φ ≺≻ σ and D(ρ) ∩ D(σ) ≠ ∅, we have σ ≃ ρ by the Subform Theorem 22.5. Thus the general Pfister form ρ is uniquely determined by φ up to isomorphism. We call ρ the associated general Pfister form of φ. If φ represents one, then ρ is a Pfister form.

24. Linkage of quadratic forms

In this section, we look at the quadratic analogue of linkage of bilinear Pfister forms. The Hauptsatz shows that anisotropic forms in I_{q}^{n}(F) have dimension at least 2^{n}. We shall be interested in those dimensions that are realizable by anisotropic forms in I_{q}^{n}(F). In this section, we determine the possible dimension of anisotropic forms that are the sum of two general quadratic Pfister forms as well as the meaning of when the sum of three general n-fold Pfister forms is congruent to zero mod I_{q}^{n}(F). We shall return to and expand these results in §35 and §82.

Proposition 24.1. Let φ ∈ GP (F).

(1) Let ρ ∈ GP_{n}(F) be a subform of φ with n ≥ 1. Then there is a bilinear Pfister form 𝔟 such that φ ≃ 𝔟 ⊗ ρ.

(2) Let 𝔟 be a general bilinear Pfister form such that φ_{𝔟} is a subform of φ. Then there is ρ ∈ P (F) such that φ ≃ 𝔟 ⊗ ρ.

Proof. We may assume that φ is anisotropic of dimension ≥ 2.

(1): Let 𝔟 be a bilinear Pfister form of the largest dimension such that 𝔟 ⊗ ρ is isometric to a subform ψ of φ. As 𝔟 ⊗ ρ in nondegenerate, V_{ψ}^{⊥} ∩ V_{ψ} = 0. We claim that ψ = φ. Suppose not. Then V_{ψ}^{⊥} ≠ 0, hence V_{ψ}^{⊥} \ V_{ψ} ≠ ∅. Choose a = −ψ(v) with v ∈ V_{ψ}^{⊥} \ V_{ψ}. Lemma 23.1 implies that ⟨⟨a⟩⟩ ⊗ ρ is isometric to a subform of φ, contradicting the maximality of 𝔟.

(2): We may assume that char F = 2 and 𝔟 is a Pfister form, so 1 ∈ D(φ_{𝔟}) ⊂ D(φ). Let W be a subspace of V_{φ} such that φ|_{W} ≃ φ_{𝔟}. Choose a vector w ∈ W such that φ_{𝔟}(w) = 1 and write the quasi-Pfister form φ_{𝔟} = ⟨1⟩ ⊥ φ′_{𝔟} where V_{φ′_{𝔟}} is any complementary subspace of F w in V_{φ_{𝔟}}. Let v ∈ V_{φ} satisfy v is orthogonal to V_{φ′_{𝔟}}, but 𝔟(v, w) ≠ 0. Then the restriction of φ on W ⊕ F v is isometric to ψ := φ′_{𝔟} ⊥ [1, a] for some a ∈ F^{×}. Note that ψ is isometric to subforms of both of the general Pfister forms φ and μ := 𝔟 ⊗ ⟨⟨a]]. In particular, ψ and μ are anisotropic. As dim ψ > \frac{1}{2} dim μ, the form ψ is a Pfister neighbor of μ. Hence ψ ≺≻ μ by Remark 23.11. Since φ_{F (ψ)} is hyperbolic by Proposition 22.18 so is φ_{F (μ)}. It follows from the Subform Theorem 22.5 that μ is isomorphic to a subform of φ as 1 ∈ D(μ) ∩ D(φ). By the first statement of the proposition, there is a bilinear Pfister form 𝔠 such that φ ≃ 𝔠 ⊗ μ = 𝔠 ⊗ 𝔟 ⊗ ⟨⟨a]]. Hence φ ≃ 𝔟 ⊗ ρ where ρ = 𝔠 ⊗ ⟨⟨a]]. □

Let ρ be a general quadratic Pfister form. We say a general quadratic Pfister form ψ (respectively, a general bilinear Pfister form 𝔟) is a divisor ρ if ρ ≃ 𝔠 ⊗ ψ for some bilinear Pfister form 𝔠 (respectively, ρ ≃ 𝔟⊗μ for some quadratic Pfister form μ). By Proposition 24.1, any general quadratic Pfister subform of ρ is a divisor of ρ and any general bilinear Pfister form 𝔟 of ρ whose associated quadratic form is a subform of ρ is a divisor ρ.

Theorem 24.2. Let φ_{1}, φ_{2} ∈ GP (F) be anisotropic. Let ρ ∈ GP (F) be a form of largest dimension such that ρ is isometric to subforms of φ_{1} and φ_{2}. Then

𝔦_{0}(φ_{1} ⊥ −φ_{2}) = dim ρ.

Proof. Note that 𝔦_{0} := 𝔦_{0}(φ_{1} ⊥ −φ_{2}) ≥ d := dim ρ. We may assume that 𝔦_{0} > 1. We claim that φ_{1} and φ_{2} have isometric nondegenerate binary subforms. To prove the claim let W be a 2-dimensional totally isotropic subspace of V_{φ_{1}}⊕V_{−φ_{2}}. As φ_{1} and φ_{2} are anisotropic, the projections U_{1} and U_{2} of W to V_{φ_{1}} and V_{−φ_{2}} = V_{φ_{2}}, respectively, are 2-dimensional. Moreover, the binary forms ψ_{1} := φ_{1}|_{U_{1}} and ψ_{2} = φ_{2}|_{U_{2}} are isometric. We may assume that ψ_{1} and ψ_{2} are degenerate (and therefore, char(F) = 2). Hence ψ_{1} and ψ_{2} are isometric to φ_{𝔟}, where 𝔟 is a 1-fold general bilinear Pfister form. By Proposition 24.1(2), we have φ_{1} ≃ 𝔟 ⊗ ρ_{1} and φ_{2} ≃ 𝔟 ⊗ ρ_{2} for some ρ_{i} ∈ P (F). Write ρ_{i} = 𝔠_{i} ⊗ ν_{i} for bilinear Pfister forms 𝔠_{i} and 1-fold quadratic Pfister forms ν_{i}. Consider quaternion algebras Q_{1} and Q_{2} whose reduced norm forms are similar to 𝔟 ⊗ ν_{1} and 𝔟 ⊗ ν_{2}, respectively. The algebras Q_{1} and Q_{2} are split by a quadratic field extension that splits 𝔟. By Theorem 98.19, the algebras Q_{1} and Q_{2} have subfields isomorphic to a separable quadratic extension L/F. By Example 9.7, the reduced norm forms of Q_{1} and Q_{2} are divisible by the nondegenerate norm form of L/F. Hence the forms 𝔟 ⊗ ν_{1} and 𝔟 ⊗ ν_{2} and therefore φ_{1} and φ_{2} have isometric nondegenerate binary subforms. The claim is proven.

By the claim, ρ is a general r-fold Pfister form with r ≥ 1. Write φ_{1} = ρ ⊥ ψ_{1} and φ_{2} = ρ ⊥ ψ_{2} for some forms ψ_{1} and ψ_{2}. We have φ_{1} ⊥ (−φ_{2}) ≃ ψ_{1} ⊥ (−ψ_{2}) ⊥ dℍ. Assume that 𝔦_{0} > d. Then the form ψ_{1} ⊥ (−ψ_{2}) is isotropic, i.e., ψ_{1} and ψ_{2} have a common value, say a ∈ F^{×}. By Lemma 23.1, the form ⟨⟨−a⟩⟩⊗ρ is isometric to subforms of φ_{1} and φ_{2}, a contradiction. □

Corollary 24.3. Let φ_{1}, φ_{2} ∈ GP_{n}(F) be anisotropic forms. Then the possible values of 𝔦_{0}(φ_{1} ⊥ −φ_{2}) are 0, 1, 2, 4, …, 2^{n}.

Let φ_{1} ∈ GP_{m}(F) and φ_{2} ∈ GP_{n}(F) be anisotropic forms satisfying 𝔦(φ_{1} ⊥ −φ_{2}) = 2^{r} > 0. Let ρ be a general quadratic r-fold Pfister form isometric to a subform of φ_{1} and to a subform of φ_{2}. We call ρ the linkage of φ_{1} and φ_{2} and say that φ_{1} and φ_{2} are r-linked. By Proposition 24.1, the linkage ρ is a divisor of φ_{1} and φ_{2}. If m = n and r ≥ n − 1, we say that φ_{1} and φ_{2} are linked.

25. THE SUBMODULE J_{n}(F) 103

Remark 24.4. Let φ_{1} and φ_{2} be general quadratic Pfister forms. Suppose that φ_{1} and φ_{2} have isometric r-fold quasi-Pfister subforms. Then 𝔦_{0}(φ_{1} ⊥ −φ_{2}) ≥ 2^{r} and by Theorem 24.2, the forms φ_{1} and φ_{2} have isometric general quadratic r-fold Pfister subforms.

For three n-fold Pfister forms, we have:

Proposition 24.5. Let φ_{1}, φ_{2}, φ_{3} ∈ P_{n}(F). If φ_{1} + φ_{2} + φ_{3} ∈ I_{q}^{n+1}(F) then there exist a quadratic (n − 1)-fold Pfister form ρ and a_{1}, a_{2}, a_{3} ∈ F^{×} such that a_{1}a_{2}a_{3} = 1 and φ_{i} ≃ ⟨⟨a_{i}⟩⟩ ⊗ ρ for i = 1, 2, 3. In particular, ρ is a common divisor of φ_{i} for i = 1, 2, 3.

Proof. We may assume that all φ_{i} are anisotropic Pfister forms by Corollary 9.10. In addition, we have (φ_{3})_{F (φ_{3})} is hyperbolic. By Proposition 23.9, the form (φ_{1} ⊥ −φ_{2})_{F (φ_{3})} is also hyperbolic. As φ_{3} is anisotropic, φ_{1} ⊥ −φ_{2} cannot be hyperbolic by the Hauptsatz 23.7. Consequently,

(φ_{1} ⊥ −φ_{2})_{an} ≃ aφ_{3} ⊥ τ

over F for some a ∈ F^{×} and a quadratic form τ by the Subform Theorem 22.5 and Proposition 7.22. As dim τ < 2^{n} and τ ∈ I_{q}^{n+1}(F), the form τ is hyperbolic by Hauptsatz 23.7 and therefore φ_{1} − φ_{2} = aφ_{3} in I_{q}(F). It follows that 𝔦_{0}(φ_{1} ⊥ −φ_{2}) = 2^{n−1}, hence φ_{1} and φ_{2} are linked by Theorem 24.2.

Let ρ be a linkage of φ_{1} and φ_{2}. By Proposition 24.1, we have φ_{1} ≃ ⟨⟨a_{1}⟩⟩ ⊗ ρ and φ_{2} ≃ ⟨⟨a_{2}⟩⟩ ⊗ ρ for some a_{1}, a_{2} ∈ F^{×}. Then φ_{3} is similar to (φ_{1} ⊥ −φ_{2})_{an} ≃ −a_{1}⟨⟨a_{1}a_{2}⟩⟩ ⊗ ρ, i.e., φ_{3} ≃ ⟨⟨a_{1}a_{2}⟩⟩ ⊗ ρ. □

Corollary 24.6. Let φ_{1}, φ_{2}, φ_{3} ∈ P_{n}(F). Suppose that

(24.7)  φ_{1} + φ_{2} + φ_{3} ≡ 0 mod I_{q}^{n+1}(F).

Then

e_{n}(φ_{1}) + e_{n}(φ_{2}) + e_{n}(φ_{3}) = 0 in H^{n}(F).

Proof. By Proposition 24.5, we have φ_{i} ≃ ⟨⟨a_{i}⟩⟩ ⊗ ρ for some ρ ∈ P_{n−1}(F) and a_{i} ∈ F^{×} for i = 1, 2, 3 satisfying a_{1}a_{2}a_{3} = 1. It follows from Proposition 16.1 that

e_{n}(φ_{1}) + e_{n}(φ_{2}) + e_{n}(φ_{3}) = e_{n}(⟨⟨a_{1}⟩⟩ ⊗ ρ) + e_{n}(⟨⟨a_{2}⟩⟩ ⊗ ρ) + e_{n}(⟨⟨a_{3}⟩⟩ ⊗ ρ)
= {a_{1}a_{2}a_{3}}e_{n−1}(ρ) = 0.  □

25. The submodule J_{n}(F)

By Corollary 23.4, a general quadratic Pfister form has the following “intrinsic” characterization: a nondegenerate anisotropic quadratic form φ of positive even dimension is a general quadratic Pfister form if and only if the form φ_{F (φ)} is hyperbolic. We shall use this to characterize elements of I_{q}^{n}(F). Let φ be a form that is nonzero in I_{q}(F). There exists a field extension K/F such that (φ_{K})_{an} is a general quadratic n-fold Pfister form for some n ≥ 1. The smallest possible such n is called the degree deg(φ) of φ. We shall see in Theorem 40.10 that φ ∈ I_{q}^{n}(F) if and only if deg φ ≥ n. In this section, we shall begin the study of the degree of forms. The ideas in this section are due to Knebusch (cf. [83] and [84]).

We begin by constructing a tower of field extensions of F with (φ_{K})_{an} a general quadratic n-fold Pfister form where K is the penultimate field K in the tower.

Let φ be a nondegenerate quadratic form over F. We construct a tower of fields F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} and quadratic forms φ_{k} over F_{k} for all k ∈ [0, h] as follows: We start with F_{0} := F, φ_{0} := φ_{an}, and inductively set F_{k} := F_{k−1}(φ_{k−1}), φ_{k} := (φ_{F_{k}})_{an} for k > 0. We stop at F_{h} such that dim φ_{h} ≤ 1. The form φ_{k} is called the kth (anisotropic) kernel form of φ. The tower of the fields F_{k} is called the generic splitting tower of φ. The integer h is called the height of φ and denoted by 𝔥(φ). We have 𝔥(φ) = 0 if and only if dim φ_{an} ≤ 1.

Let h = 𝔥(φ). For any k ∈ [0, h], the k-th absolute higher Witt index 𝔧_{k}(φ) of φ is defined as the integer 𝔦_{0}(φ_{F_{k}}). Clearly, one has

0 ≤ 𝔧_{0}(φ) < 𝔧_{1}(φ) < ⋯ < 𝔧_{h}(φ) = [(dim φ)/2].

The set of integers {𝔧_{0}(φ), …, 𝔧_{h}(φ)} is called the splitting pattern of φ.

Proposition 25.1. Let φ be a nondegenerate quadratic form with h = 𝔥(φ). The splitting pattern {𝔧_{0}(φ), …, 𝔧_{h}(φ)} of φ coincides with the set of Witt indices 𝔦_{0}(φ_{K}) over all field extensions K/F.

Proof. Let K/F be a field extension. Define a tower of fields K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{h} by K_{0} = K and K_{k} = K_{k−1}(φ_{k−1}) for k > 0. Clearly, F_{k} ⊂ K_{k} for all k. Let k ≥ 0 be the smallest integer such that φ_{k} is anisotropic over K_{k}. It suffices to show that 𝔦_{0}(φ_{K}) = 𝔧_{k}(φ).

By definition of φ_{k} and 𝔧_{k}, we have φ_{F_{k}} = φ_{k} ⊥ 𝔧_{k}(φ)ℍ. Therefore, φ_{K_{k}} = (φ_{q})_{K_{k}} ⊥ 𝔧_{k}(φ)ℍ. As φ_{k} is anisotropic over K_{k}, we have 𝔦_{0}(φ_{K_{k}}) = 𝔧_{k}(φ).

We claim that the extension K_{k}/K is purely transcendental. This is clear if k = 0. Otherwise K_{k} = K_{k−1}(φ_{k−1}) is purely transcendental by Proposition 22.9, since φ_{k−1} is isotropic over K_{k−1} by the choice of k and is nondegenerate. It follows from the claim and Remark 8.9 that 𝔦_{0}(φ_{K}) = 𝔦_{0}(φ_{K_{k}}) = 𝔧_{k}(φ). □

Corollary 25.2. Let φ be a nondegenerate quadratic form over F and let K/F be a purely transcendental extension. Then the splitting patterns of φ and φ_{K} are the same.

Proof. This follows from Lemma 7.15. □

We define the relative higher Witt index 𝔦_{k}(φ), k ∈ [1, 𝔥(φ)], of a nondegenerate quadratic form φ to be the difference

𝔦_{k}(φ) = 𝔧_{k}(φ) − 𝔧_{k−1}(φ).

Clearly, 𝔦_{k}(φ) > 0 and 𝔦_{k}(φ) = 𝔦_{r}(φ_{s}) for any r > 0 and s ≥ 0 such that r + s = k.

Corollary 25.3. Let φ be a nondegenerate anisotropic quadratic form over F of dimension at least two. Then

𝔦_{1}(φ) = 𝔧_{1}(φ) = min{𝔦_{0}(φ_{K}) | K/F a field extension with φ_{K} isotropic}.

Let φ be a nondegenerate nonhyperbolic quadratic form of even dimension over F with h = 𝔥(φ). Let F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} be the generic splitting tower of φ. The form φ_{h−1} = (φ_{F_{h−1}})_{an} is hyperbolic over its function field, hence a general n-fold Pfister form for some integer n ≥ 1 with 𝔦_{h}(φ) = 2^{n−1} by Corollary 23.4. The form φ_{h−1} is called the leading form of φ and n is called the degree of φ and is denoted by deg φ. The field F_{h−1} is called the leading field of φ. For convenience, we set deg φ = ∞ if φ is hyperbolic.

Remark 25.4. Let φ be a nondegenerate quadratic form of even dimension with the generic splitting tower F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h}. If φ_{k} = (φ_{F_{k}})_{an} with k ∈ [0, h−1], then deg φ_{k} = deg φ.

Notation 25.5. Let φ be a nondegenerate quadratic form over F and X = X_{φ}. Let k ∈ [0, 𝔥(φ)]. We shall let X_{k} := X_{φ_{k}} and also write 𝔧_{k}(X) (respectively, 𝔦_{k}(X)) for 𝔧_{k}(φ) (respectively, 𝔦_{k}(φ)).

It is a natural problem to classify nondegenerate quadratic forms over a field F of a given height. This is still an open problem even for forms of height two. By Corollary 23.4, we do know

Proposition 25.6. Let φ be an even-dimensional nondegenerate anisotropic quadratic form. Then 𝔥(φ) = 1 if and only if φ ∈ GP (F).

Proposition 25.7. Let φ be a nondegenerate quadratic form of even dimension over F and K/F a field extension with (φ_{K})_{an} an m-fold general Pfister for some m ≥ 1. Then m ≥ deg φ. In particular, deg φ is the smallest integer n ≥ 1 such that (φ_{K})_{an} is a general n-fold Pfister form over an extension K/F.

Proof. It follows from Proposition 25.1 that

(dim φ − 2^{m})/2 = 𝔦_{0}(φ_{K}) ≤ 𝔧_{𝔥(φ)−1}(φ) = (dim φ − 2^{deg φ})/2,

hence the inequality. □

Corollary 25.8. Let φ be a nondegenerate quadratic form of even dimension over F. Then deg φ_{E} ≥ deg φ for any field extension E/F.

For every n ≥ 1 set

J_{n}(F) := {φ ∈ I_{q}(F) | deg φ ≥ n} ⊂ I_{q}(F).

Clearly, J_{1}(F) = I_{q}(F).

Lemma 25.9. Let ρ ∈ GP_{n}(F) be anisotropic with n ≥ 1. Let φ ∈ J_{n+1}(F). Then deg(ρ ⊥ φ) ≤ n.

Proof. We may assume that φ is not hyperbolic. Let ψ = ρ ⊥ φ. Let F_{0}, F_{1}, …, F_{h} be the generic splitting tower of φ and let φ_{i} = (φ_{F_{i}})_{an}. We show that ρ_{F_{h}} is anisotropic. Suppose not. Choose j maximal such that ρ_{F_{j}} is anisotropic. Then ρ_{F_{j+1}} is hyperbolic so dim φ_{j} ≤ dim ρ by the Subform Theorem 22.5. Hence

2^{n} = dim ρ ≥ dim φ_{j} ≥ deg 2^{deg φ_{j}} = 2^{deg φ} ≥ 2^{n+1}

which is impossible. Thus ρ_{F_{𝔥}} is anisotropic.

As φ is hyperbolic over F_{h}, we have ψ_{F_{h}} ∼ ρ_{F_{h}}. Consequently,

deg ψ ≤ deg ψ_{F_{h}} = deg ρ_{F_{h}} = n,

hence deg ψ ≤ n as claimed. □

Corollary 25.10. Let φ and ψ be any even-dimensional nondegenerate quadratic forms. Then deg(φ ⊥ ψ) ≥ min(deg φ, deg ψ).

Proof. If either φ or ψ is hyperbolic, this is trivial, so assume that both forms are not hyperbolic. We may also assume that φ ⊥ ψ is not hyperbolic. Let K/F be a field extension such that (φ ⊥ ψ)_{K} ∼ ρ for some ρ ∈ GP_{n}(K) where n = deg(φ ⊥ ψ). Then φ_{K} ∼ ρ ⊥ (−ψ_{K}). Suppose that deg ψ > n. Then deg ψ_{K} > n and applying the lemma to the form ρ ⊥ (−ψ_{K}) implies deg φ_{K} ≤ n. Hence deg φ ≤ n = deg(φ ⊥ ψ). □

Proposition 25.11. J_{n}(F) is a W (F)-submodule of I_{q}(F) for every n ≥ 1.

Proof. Corollary 25.10 shows that J_{n}(F) is a subgroup of I_{q}(F). Since deg φ = deg(aφ) for all a ∈ F^{×}, it follows that J_{n}(F) is also closed under multiplication by elements of W (F). □

Corollary 25.12. I_{q}^{n}(F) ⊂ J_{n}(F).

Proof. As general quadratic n-fold Pfister forms clearly lie in J_{n}(F), the result follows from Proposition 25.11. □

Proposition 25.13. I_{q}^{2}(F) = J_{2}(F).

Proof. Let φ ∈ J_{2}(F) and φ_{k} = φ_{F_{k}} with F_{k}, k ∈ [0, h], the generic splitting tower. As deg φ ≥ 2, the field F_{k} is the function field of a smooth quadric of dimension at least 2 over F_{k−1}, hence the field F_{k−1} is algebraically closed in F_{k}. Since the form φ_{h} = 0 has trivial discriminant, by descending induction on k, it follows that φ = φ_{0} has trivial discriminant. By Theorem 13.7 we conclude that φ ∈ I_{q}^{2}(F). □

Proposition 25.14. J_{3}(F) = {φ | dim φ is even, disc(φ) = 1, clif(φ) = 1}.

Proof. Let φ be an anisotropic form of even dimension and trivial discriminant. Then φ ∈ I_{q}^{2}(F) = J_{2}(F) by Theorem 13.7 and Proposition 25.13. Suppose φ also has trivial Clifford invariant. We must show that deg φ ≥ 3. Let K be the leading field of φ and ρ its leading form. Then ρ ∈ GP_{n}(F) with n ≥ 2. Suppose that n = 2. As e_{2}(ρ) = 0 in H^{2}(K), we have ρ is hyperbolic by Corollary 12.5, a contradiction. Therefore, φ ∈ J_{3}(F).

Let φ ∈ J_{3}(F). Then φ ∈ I_{q}^{2}(F) by Proposition 25.13. In particular, disc(φ) = 1 and φ = ∑_{i=1}^{r} ρ_{i} with ρ_{i} ∈ GP_{2}(F), 1 ≤ i ≤ r. We show that clif(φ) = 1 by induction on r. Let ρ_{r} = b⟨⟨a, d]] and K = F_{d}. Then φ_{K} ∈ J_{3}(K) and satisfies φ_{K} = ∑_{i=1}^{r−1}(ρ_{i})_{K} as (ρ_{r})_{K} is hyperbolic. By induction, clif(φ_{K}) = 1. Thus clif(φ) lies in kernel of Br(F) → Br(K). Therefore, the index of clif(φ) is at most two. Consequently, clif(φ) is represented by a quaternion algebra, hence there exists a 2-fold quadratic Pfister form σ satisfying clif(φ) = clif(σ). Thus clif(φ + σ) = 1, so φ + σ lies in J_{3}(F) by the first part of the proof. It follows that σ lies in J_{3}(F). Therefore, σ = 0 and clif(φ) = 1. □

As ē_{2} is an isomorphism (cf. Theorem 16.3 if char F = 2 and Chapter VIII if char F ≠ 2), we have I^{3}(F) = J_{3}(F). We shall show that I^{n}(F) = J_{n}(F) for all n in Theorem 40.10. In this section, we show the following due to Arason and Knebusch (cf. [9]).

Proposition 25.15. I^{m}(F)J_{n}(F) ⊂ J_{n+m}(F).

Proof. Clearly, it suffices to do the case that m = 1. Since 1-fold bilinear Pfister forms additively generate I(F), it also suffices to show that if φ ∈ J_{n}(F) and a ∈ F^{×}, then ⟨⟨a⟩⟩ ⊗ φ ∈ J_{n+1}(F). Let ψ be the anisotropic part of ⟨⟨a⟩⟩ ⊗ φ. We may assume that ψ ≠ 0.

First suppose that ψ ∈ GP (F). We prove that deg ψ > n by induction on the height h of φ. If h = 1, then φ ∈ GP (F) and the result is clear. So assume that h > 1. Suppose that ψ_{F (φ)} remains anisotropic. Applying the induction hypothesis to the form φ_{F (φ)}, we have

deg ψ = deg ψ_{F (φ)} > n.

If ψ_{F (φ)} is isotropic, it is hyperbolic and therefore dim ψ ≥ dim φ by the Subform Theorem 22.5. As h > 1, we have

2^{deg ψ} = dim ψ ≥ dim φ > 2^{deg φ} ≥ 2^{n},

hence deg ψ > n.

Now consider the general case. Let K/F be a field extension such that ψ_{K} is Witt-equivalent to a general Pfister form and deg ψ_{K} = deg ψ. By the first part of the proof

deg ψ = deg ψ_{K} > n.  □

26. The Separation Theorem

There are anisotropic quadratic forms φ and ψ with dim φ < dim ψ and φ_{F (ψ)} isotropic. For example, this is the case when φ and ψ are Pfister neighbors of the same Pfister form. In this section, we show that if two anisotropic quadratic forms φ and ψ are separated by a power of two, more precisely, if dim φ ≤ 2^{n} < dim ψ for some n ≥ 0, then φ_{F (ψ)} remains anisotropic.

We shall need the following observation.

Remark 26.1. Let ψ be a quadratic form. Then V_{ψ} contains a (maximal) totally isotropic subspace of dimension 𝔦′_{0}(ψ) := 𝔦_{0}(ψ) + dim(rad ψ). Define the invariant s of a form by s(ψ) := dim ψ − 2𝔦′_{0}(ψ) = dim ψ_{an} − dim(rad ψ). If two quadratic forms ψ and μ are Witt-equivalent, then s(ψ) = s(μ).

A field extension L/F is called unirational if there is a field extension L′/L with L′/F purely transcendental. A tower of unirational field extensions is unirational. If L/F is unirational, then every anisotropic quadratic form over F remains anisotropic over L by Lemma 7.15.

Lemma 26.2. Let φ be an anisotropic quadratic form over F satisfying dim φ ≤ 2^{n} for some n ≥ 0. Then there exists a field extension K/F and an (n + 1)-fold anisotropic quadratic Pfister form ρ over K such that

(1) φ_{K} is isometric to a subform of ρ.

(2) The field extension K(ρ)/F is unirational.

Proof. Let K_{0} = F (t_{1}, …, t_{n+1}) and let ρ = ⟨⟨t_{1}, …, t_{n+1}]]. Then ρ is anisotropic. Indeed, by Corollary 19.6 and induction, it suffices to show ⟨⟨t]] is anisotropic over F (t). If this is false, there is an equation f^{2} + fg + tg^{2} = 0 with f, g ∈ F [t]. Looking at the highest term of t in this equation gives either a^{2}t^{2n} = 0 or b^{2}t^{2n+1} = 0 where a, b are the leading coefficients of f, g, respectively. Neither is possible.

Consider the class F of field extensions E/K_{0} satisfying:

(1′) ρ is anisotropic over E.

(2′) The field extension E(ρ)/F is unirational.

We show that K_{0} ∈ F. By the above ρ is anisotropic. Let L = K_{0}(⟨⟨1, t_{1}]]). Then L/F is purely transcendental. As ρ_{L} is isotropic, L(ρ)/L is also purely transcendental and hence so is L(ρ)/F. Since K_{0}(ρ) ⊂ L(ρ), the field extension K_{0}(ρ)/F is unirational.

For every field E ∈ F, the form φ_{E} is anisotropic by (2′). As ρ_{E} is nondegenerate, the form ρ_{E} ⊥ (−φ_{E}) is regular. We set

m(E) = 𝔦_{0}(ρ_{E} ⊥ (−φ_{E})) = 𝔦′_{0}(ρ_{E} ⊥ (−φ_{E}))

and let m be the maximum of the m(E) over all E ∈ F.

Claim 1: We have m(E) ≤ dim φ and if m(E) = dim φ, then φ_{E} is isometric to a subform of ρ_{E}.

Let W be a totally isotropic subspace in V_{ρ_{E}} ⊥ V_{−φ_{E}} of dimension m(E). Since ρ_{E} and φ_{E} are anisotropic, the projections of W to V_{ρ_{E}} and V_{−φ_{E}} = V_{φ_{E}} are injective. This gives the inequality. Suppose that m(E) = dim φ. Then the projection p : W → V_{φ_{E}} is an isomorphism and the composition

V_{φ_{E}} ⟶^{p^{−1}} W → V_{ρ_{E}}

identifies φ_{E} with a subform of ρ_{E}.

Claim 2: m = dim φ.

Assume that m < dim φ. We derive a contradiction. Let K ∈ F be a field satisfying m = m(K) and set τ = (ρ_{K} ⊥ (−φ_{K}))_{an}. As the form ρ_{K} ⊥ (−φ_{K}) is regular, we have τ ∼ ρ_{K} ⊥ (−φ_{K}) and

(26.3)  dim ρ + dim φ = dim τ + 2m.

Let W be a totally isotropic subspace in V_{ρ_{K}} ⊥ V_{−φ_{K}} of dimension m. Let σ denote the restriction of ρ_{K} on V_{ρ_{K}} ∩ W^{⊥}. Thus σ is a subform of ρ_{K} of dimension ≥ 2^{n+1} − m > 2^{n}. In particular, σ is a Pfister neighbor of ρ_{K}. By Lemma 8.10, the natural map V_{ρ_{K}} ∩ W^{⊥} → W^{⊥}/W identifies σ with a subform of τ.

We show that condition (2′) holds for K(τ). Since σ is a Pfister neighbor of ρ_{K}, the form σ and therefore τ is isotropic over K(ρ). By Lemma 22.14, the extension K(ρ)/K is separable, hence τ_{K(ρ)} is regular by Lemma 22.13. Therefore, by Lemma 22.9 the extension K(ρ)(τ)/K(ρ) is purely transcendental. It follows that K(ρ)(τ) = K(τ)(ρ) is unirational over F, hence condition (2′) is satisfied.

As τ is isotropic over K(τ), we have m(K(τ)) > m, hence K(τ) ∉ F. Therefore, condition (1′) does not hold for K(τ), i.e., ρ_{K} is isotropic and therefore hyperbolic over K(τ). As ∅ ≠ D(σ) ⊂ D(ρ_{K}) ∩ D(τ), the form τ is isometric to a subform of ρ_{K} by the Subform Theorem 22.5. Let τ^{⊥} be the complementary form of τ in ρ_{K}. It follows from (26.3) that

dim τ^{⊥} = dim ρ − dim τ = 2m − dim φ < dim φ.

As ρ_{K} ⊥ (−τ) ∼ τ^{⊥} by Lemma 8.13,

(26.4)  τ ⊥ (−τ) ∼ ρ_{K} ⊥ (−φ_{K}) ⊥ (−τ) ∼ τ^{⊥} ⊥ (−φ_{K}).

We now use the invariant s defined in Remark 26.1. Since the space of τ ⊥ (−τ) contains a totally isotropic subspace of dimension dim τ, it follows from (26.4) and Remark 26.1 that

s(τ^{⊥} ⊥ (−φ_{K})) = s(τ ⊥ (−τ)) = 0,

i.e., the form τ^{⊥} ⊥ (−φ_{K}) contains a totally isotropic subspace of half the dimension of the form. Since dim φ > dim τ^{⊥}, this subspace intersects V_{φ_{K}} nontrivially, consequently φ_{K} is isotropic contradicting condition (2′). This establishes the claim.

It follows from the claims that φ_{K} is isometric to a subform of ρ_{K}. □

Theorem 26.5 (Separation Theorem). Let φ and ψ be two anisotropic quadratic forms over F. Suppose that dim φ ≤ 2^{n} < dim ψ for some n ≥ 0. Then φ_{F (ψ)} is anisotropic.

Proof. Let ρ be an (n + 1)-fold Pfister form over a field extension K/F as in Lemma 26.2 with φ_{K} a subform of ρ. By the lemma ψ_{K(ρ)} is anisotropic. Suppose that φ_{K(ψ)} is isotropic. Then ρ_{K(ψ)} is isotropic, hence hyperbolic. By the Subform Theorem 22.5, there exists an a ∈ F such that aψ_{K} is a subform of ρ. As dim ψ > \frac{1}{2} dim ρ, the form aψ_{K} is a neighbor of ρ, hence aψ_{K(ρ)} and therefore ψ_{K(ρ)} is isotropic. This is a contradiction. □

Corollary 26.6. Let φ and ψ be two anisotropic quadratic forms over F with dim ψ ≥ 2. If dim ψ ≥ 2 dim φ − 1, then φ_{F (ψ)} is anisotropic.

The Separation Theorem was first proved by Hoffmann in [53] for fields of characteristic different from 2 and by Hoffmann and Laghribi in [58] for fields of characteristic 2. We shall give another, more geometric proof, in 79.8 below.

27. A further characterization of quadratic Pfister forms

In this section, we give a further characterization of quadratic Pfister forms due to Fitzgerald (cf. [44]) used to answer a question of Knebusch in [84] for fields of characteristic different from 2. Knebusch and Scharlau showed in [85] that Fitzgerald’s theorem implied that if a nondegenerate anisotropic quadratic form ρ becomes hyperbolic over the function field of an irreducible anisotropic form φ satisfying dim φ > \frac{1}{3} dim ρ, then ρ is a general quadratic Pfister form. Hoffmann and Laghribi showed that this characterization was also true for fields of characteristic 2 in [57].

For a nondegenerate nonhyperbolic quadratic form ρ of even dimension, we set N(ρ) = dim ρ − 2^{deg ρ}. Since the splitting patterns of ρ and ρ_{F (t)} are the same, by Corollary 25.2, we have N(ρ_{F (t)}) = N(ρ).

Theorem 27.1. Let ρ be a nonhyperbolic quadratic form and φ a subform of ρ of dimension at least 2. Suppose that:

(1) φ and its complementary form in ρ are anisotropic.

(2) ρ_{F (φ)} is hyperbolic.

(3) 2 dim φ > N(ρ). Then ρ is an anisotropic general Pfister form.

Proof. Note that ρ is a nondegenerate form of even dimension by Remark 7.18 as ρ_{F (φ)} is hyperbolic.

Claim 1: For any field extension K/F with φ_{K} anisotropic and ρ_{K} not hyperbolic, φ_{K} is isometric to a subform of (ρ_{K})_{an}.

By Lemma 8.13, the form ρ ⊥ (−φ) is Witt-equivalent to ψ := φ^{⊥}. In particular, dim ρ = dim φ + dim ψ. Set ρ′ = (ρ_{K})_{an}. It follows from (3) that

dim(ρ′ ⊥ (−φ_{K})) ≥ 2^{deg ρ} + dim φ > dim ρ − dim φ = dim ψ.

As ρ′ ⊥ (−φ_{K}) ∼ ψ_{K} it follows that the form ρ′ ⊥ (−φ_{K}) is isotropic, therefore D(ρ′) ∩ D(φ_{K}) ≠ ∅. Since ρ′_{K(φ)} is hyperbolic, the form φ_{K} is isometric to a subform of ρ′ by the Subform Theorem 22.5 as needed.

Claim 2: ρ is anisotropic.

Applying Claim 1 to K = F implies that φ is isometric to a subform of ρ′ = ρ_{an}. Let ψ′ be the complementary form of φ in ρ′. By Lemma 8.13,

ψ′ ∼ ρ′ ⊥ (−φ) ∼ ρ ⊥ (−φ) ∼ ψ.

As both forms ψ and ψ′ are anisotropic, we have ψ′ ≃ ψ. Hence

dim ρ = dim φ + dim ψ = dim φ + dim ψ′ = dim ρ′ = dim ρ_{an}.

Therefore, ρ is anisotropic.

We now investigate the form φ_{F (ρ)}. Suppose it is isotropic. Then φ ≺≻ ρ, hence ρ_{F (ρ)} is hyperbolic by Proposition 22.18. It follows that ρ is a general Pfister form by Corollary 23.4 and we are done. Thus we may assume that φ_{F (ρ)} is anisotropic. Normalizing we may also assume that 1 ∈ D(φ). We shall prove that ρ is a Pfister form by induction on dim ρ. Suppose that ρ is not a Pfister form. In particular, ρ_{1} := (ρ_{F (ρ)})_{an} is nonzero and dim ρ_{1} ≥ 2. We shall finish the proof by obtaining a contradiction. Let φ_{1} = φ_{F (ρ)}.

Note that deg ρ_{1} = deg ρ and dim ρ_{1} < dim ρ, hence N(ρ_{1}) < N(ρ).

Claim 3: ρ_{1} is a Pfister form.

Applying Claim 1 to the field K = F (ρ), we see that φ_{1} is isometric to a subform of ρ_{1}. We have

2 dim φ_{1} = 2 dim φ > N(ρ) > N(ρ_{1}).

By the induction hypothesis applied to the form ρ_{1} and its subform, φ_{1}, we conclude that the form ρ_{1} is a Pfister form proving the claim. In particular, dim ρ_{1} = 2^{deg ρ_{1}} = 2^{deg ρ}.

Claim 4: D(ρ) = G(ρ).

Since G(ρ) ⊂ D(ρ), it suffices to show if x ∈ D(ρ), then x ∈ G(ρ). Suppose that x ∉ G(ρ). Hence the anisotropic part β of the isotropic form ⟨⟨x⟩⟩ ⊗ ρ is nonzero. It follows from Proposition 25.15 that deg β ≥ 1 + deg ρ.

Suppose that β_{F (ρ)} is hyperbolic. As ρ − β = −xρ in I_{q}(F), the form ρ ⊥ (−β) is isotropic, hence D(ρ) ∩ D(β) ≠ ∅. It follows from this that ρ is isometric to a subform of β by the Subform Theorem 22.5. Let β ≃ ρ ⊥ μ ∼ ρ ⊥ (−xρ) for some form μ. By Witt cancellation, μ ∼ −xρ. But dim β < 2 dim ρ, hence dim μ < dim ρ. As ρ is anisotropic, this is a contradiction. It follows that the form β_{1} = (β_{F (ρ)})_{an} is not zero and hence dim β_{1} ≥ 2^{deg β} ≥ 2^{1+deg ρ}.

Since ρ is hyperbolic over F (φ), it follows from the Subform Theorem 22.5 that φ is isometric to a subform of xρ. Applying Claim 1 to the form xρ_{F (ρ)}, we conclude that φ_{1} is a subform of xρ_{1}. As φ_{1} is also a subform of ρ_{1}, the form ⟨⟨x⟩⟩⊗ρ_{1} contains φ_{1} ⊥ (−φ_{1}) and therefore a totally isotropic subspace of dimension dim φ_{1} = dim φ. Therefore, dim(⟨⟨x⟩⟩ ⊗ ρ_{1})_{an} ≤ 2 dim ρ_{1} − 2 dim φ. Consequently,

2^{1+deg ρ} ≤ dim β_{1} = dim(⟨⟨x⟩⟩ ⊗ ρ_{1})_{an} ≤ 2 dim ρ_{1} − 2 dim φ < 2^{1+deg ρ},

a contradiction. This proves the claim.

Let F (T) = F (t_{1}, …, t_{n}) with n = dim ρ. We have deg ρ_{F (T)} = deg ρ and N(ρ_{F (T)}) = N(ρ). Working over F (T) instead of F, we have the forms φ_{F (T)} and ρ_{F (T)} satisfy the conditions of the theorem. By Claim 4, we conclude that G(ρ_{F (T)}) = D(ρ_{F (T)}). It follows from Theorem 23.2 that ρ is a Pfister form, a contradiction. □

Corollary 27.2. Let ρ be a nonzero anisotropic quadratic form and φ an irreducible anisotropic quadratic form satisfying dim φ > \frac{1}{3} dim ρ. If ρ_{F (φ)} is hyperbolic, then ρ ∈ GP (F).

Proof. As ρ_{F (φ)} is hyperbolic, the form ρ is nondegenerate. It follows by the Subform Theorem 22.5 that aφ is a subform of ρ for some a ∈ F^{×}. As ρ is anisotropic, the complementary form of aφ in ρ is anisotropic.

Let K be the leading field of ρ and τ its leading form. We show that φ_{K} is anisotropic. If φ_{F (ρ)} is isotropic, then φ ≺≻ ρ. In particular, ρ_{F (ρ)} is hyperbolic by Proposition 22.18, hence K = F and φ is anisotropic by hypothesis. Thus we may assume that φ_{F (ρ)} is anisotropic. The assertion now follows by induction on 𝔥(ρ). As τ_{K(φ)} ∼ ρ_{K(φ)} is hyperbolic, dim φ = dim φ_{K} ≤ dim τ = 2^{deg ρ} by the Subform Theorem 22.5. Hence N(ρ) = dim ρ − 2^{deg ρ} ≤ dim ρ − dim φ < 2 dim φ. The result follows by Theorem 27.1. □

The restriction dim φ > \frac{1}{3} dim ρ above is best possible (cf. [85]).

A further application of Theorem 27.1 is given by:

Theorem 27.3. Let φ and ψ be nondegenerate quadratic forms over F of the same odd dimension. If 𝔦_{0}(φ_{K}) = 𝔦_{0}(ψ_{K}) for any field extension K/F, then φ and ψ are similar.

Proof. We may assume that φ and ψ are anisotropic and have the same determinants (cf. Remark 13.8). Let n = dim φ. We shall show that φ ≃ ψ by induction on n. The statement is obvious if n = 1, so assume that n > 1.

We construct a nondegenerate form ρ of dimension 2n and trivial discriminant containing φ such that φ^{⊥} ≃ −ψ as follows: If char F ≠ 2, let ρ = φ ⊥ (−ψ). If char F = 2, write φ ≃ ⟨a⟩ ⊥ φ′ and ψ ≃ ⟨a⟩ ⊥ ψ′ for some a ∈ F^{×} and nondegenerate forms φ′ and ψ′. Set ρ = [a, c] ⊥ φ′ ⊥ ψ′, where c is chosen so that disc ρ is trivial.

By induction applied to the anisotropic parts of φ_{F (φ)} and ψ_{F (φ)}, we have φ_{F (φ)} ≃ ψ_{F (φ)}. It follows from Witt Cancellation and Proposition 13.6 (in the case char F = 2) that ρ_{F (φ)} is hyperbolic. If ρ itself is not hyperbolic, then by Theorem 27.1, the form ρ is an anisotropic general Pfister form of dimension 2n. In particular, n is a power of 2, a contradiction.

Thus ρ is hyperbolic. By Lemma 8.13, we have −φ ∼ ρ ⊥ (−φ) ∼ φ^{⊥} ≃ −ψ. As φ and ψ have the same dimension we conclude that φ ≃ ψ. □

28. Excellent quadratic forms

In general, if φ is a nondegenerate quadratic form and K/F a field extension then the anisotropic part of φ_{K} will not necessarily be isometric to a form defined over F and extended to K. Those forms over a field F whose anisotropic part over any extension field of F is defined over F are called excellent forms. These forms were first introduced by Knebusch in [84]. We study them in this section.

Let K/F be a field extension and ψ a quadratic form over K. We say that ψ is defined over F if there is a quadratic form η over F such that ψ ≃ η_{K}.

Theorem 28.1. Let φ be an anisotropic nondegenerate quadratic form of dimension ≥ 2. Then φ is a Pfister neighbor if and only if the quadratic form (φ_{F (φ)})_{an} is defined over F.

Proof. Let φ be a Pfister neighbor and let ρ be the associated general Pfister form so φ is a subform of ρ. As φ_{F (φ)} is isotropic, the general Pfister form ρ_{F (φ)} is hyperbolic by Corollary 9.10. By Lemma 8.13, the form φ_{F (φ)} is Witt-equivalent to −(φ^{⊥})_{F (φ)}. Since dim φ^{⊥} < (dim ρ)/2, it follows by Corollary 26.6 that (φ^{⊥})_{F (ρ)} is anisotropic. By Corollary 22.17, the form (φ^{⊥})_{F (φ)} is also anisotropic as φ ≺≻ ρ by Remark 23.11. Consequently, (φ_{F (φ)})_{an} ≃ (−φ^{⊥})_{F (φ)} is defined over F.

Suppose now that (φ_{F (φ)})_{an} ≃ ψ_{F (φ)} for some (anisotropic) form ψ over F. Note that dim ψ < dim φ.

Claim: There exists a form ρ satisfying:

(1) φ is a subform of ρ.

(2) The complementary form φ^{⊥} is isometric to −ψ.

(3) ρ_{F (φ)} is hyperbolic. Moreover, if dim φ ≥ 3, then ρ can be chosen in I_{q}^{2}(F).

Suppose that dim φ is even or char F ≠ 2. Then ρ = φ ⊥ (−ψ) satisfies (1), (2), and (3). As F is algebraically closed in F (φ), if dim φ ≥ 3, we have disc φ = disc ψ, hence ρ ∈ I_{q}^{2}(F).

So we may assume that char F = 2 and dim φ is odd. Write φ = φ′ ⊥ ⟨a⟩ and ψ = ψ′ ⊥ ⟨b⟩ for nondegenerate forms φ′, ψ′ and a, b ∈ F^{×}. Note that ⟨a⟩ (respectively, ⟨b⟩) is the restriction of φ (respectively, ψ) on rad 𝔟_{φ} (respectively, rad 𝔟_{ψ}) by Proposition 7.31. By definition of ψ we have ⟨a⟩_{F (φ)} ≃ ⟨b⟩_{F (φ)}. Since F (φ)/F is a separable field extension by Lemma 22.14, we have ⟨a⟩ ≃ ⟨b⟩. Therefore, we may assume that b = a.

Choose c ∈ F such that disc(φ′ ⊥ ψ′) = disc[a, c] and set ρ = φ′ ⊥ ψ′ ⊥ [a, c] so that ρ ∈ I_{q}^{2}(F). Clearly, φ is a subform of ρ and φ^{⊥} is isometric to ψ. By Lemma 8.13, ρ ⊥ φ ∼ ψ. Since φ and ψ are Witt-equivalent over F (φ), we have ρ_{F (φ)} ⊥ φ_{F (φ)} ∼ φ_{F (φ)}. Cancelling the nondegenerate form φ′_{F (φ)} yields

ρ_{F (φ)} ⊥ ⟨a⟩_{F (φ)} ∼ ⟨a⟩_{F (φ)}.

As ρ ∈ I_{q}^{2}(F) by Proposition 13.6, we have ρ_{F (φ)} ∼ 0 establishing the claim.

As dim ρ = dim φ + dim ψ < 2 dim φ and φ is anisotropic, it follows that ρ is not hyperbolic. Moreover, φ and its complement φ^{⊥} ≃ −ψ are anisotropic. Consequently, ρ is a general Pfister form by Theorem 27.1, hence φ is a Pfister neighbor. □

Exercise 28.2. Let φ be a nondegenerate quadratic form of odd dimension. Then 𝔥(φ) = 1 if and only if φ is a Pfister neighbor of dimension 2^{n} − 1 for some n ≥ 1.

Theorem 28.3. Let φ be a nondegenerate quadratic form. Then the following two conditions are equivalent:

(1) For any field extension K/F, the form (φ_{K})_{an} is defined over F.

(2) There are anisotropic Pfister neighbors φ_{0} = φ_{an}, φ_{1}, …, φ_{r−1} with associated general Pfister forms ρ_{0}, ρ_{1}, …, ρ_{r−1}, respectively, satisfying φ_{i} ≃ (ρ_{i} ⊥ φ_{i+1})_{an} for all i ∈ [0, r − 1] with a form φ_{r} of dimension at most one.

Proof. (2) ⇒ (1): Let K/F be a field extension. If all general Pfister forms ρ_{i} are hyperbolic over K, the isomorphisms in (2) show that all the φ_{i} are also hyperbolic. In particular, (φ_{K})_{an} is the zero form and hence is defined over F.

Let s be the smallest integer such that (ρ_{s})_{K} is not hyperbolic. Then the forms φ = φ_{0}, φ_{1}, …, φ_{s} are Witt-equivalent and (φ_{s})_{K} is a Pfister neighbor of the anisotropic general Pfister form (ρ_{s})_{K}. In particular, (φ_{s})_{K} is anisotropic and therefore (φ_{K})_{an} = (φ_{s})_{K} is defined over F.

(1) ⇒ (2): We prove the statement by induction on dim φ. We may assume that dim φ_{an} ≥ 2. By Theorem 28.1, the form φ_{an} is a Pfister neighbor. Let ρ be the associated general Pfister form of φ_{an}. Consider the negative of the complementary form ψ = −(φ_{an})^{⊥} of φ_{an} in ρ. It follows from Lemma 8.13 that φ_{an} ≃ (ρ ⊥ ψ)_{an}.

We claim that the form ψ satisfies (1). Let K/F be a field extension. If ρ is hyperbolic over K, then φ_{K} and ψ_{K} are Witt-equivalent. Therefore, (ψ_{K})_{an} ≃ (φ_{K})_{an} is defined over F. If ρ_{K} is anisotropic, then so is ψ_{K}, therefore (ψ_{K})_{an} = ψ_{K} is defined over F. By the induction hypothesis applied to ψ, there are anisotropic Pfister neighbors φ_{1} = ψ, φ_{2}, …, φ_{r−1} with the associated general Pfister forms ρ_{1}, ρ_{2}, …, ρ_{r−1}, respectively, such that φ_{i} ≃ (ρ_{i} ⊥ φ_{i+1})_{an} for all i ∈ [1, r − 1], where φ_{r} is a form of dimension at most 1. To finish the proof let φ_{0} = (φ)_{an} and ρ_{0} = ρ. □

A quadratic form φ satisfying equivalent conditions of Theorem 28.3 is called excellent. By Lemma 8.13, the form φ_{i+1} in Theorem 28.3(2) is isometric to the negative of the complement of φ_{i} in ρ_{i}. In particular, the sequences of forms φ_{i} and ρ_{i} are uniquely determined by φ up to isometry. Note that all forms φ_{i} are also excellent; this allows inductive proofs while working with excellent forms.

Example 28.4. If char F ≠ 2, then the form n⟨1⟩ is excellent for every n > 0.

Proposition 28.5. Let φ be an excellent quadratic form. Then in the notation of Theorem 28.3 we have the following:

(1) The integer r coincides with the height of φ.

(2) If F_{0} = F, F_{1}, …, F_{r} is the generic splitting tower of φ, then (φ_{F_{i}})_{an} ≃ (φ_{i})_{F_{i}} for all i ∈ [0, r].

Proof. The last statement is obvious if i = 0. As ρ_{0} is hyperbolic over F_{1} = F (φ_{an}) = F (φ_{0}), the forms φ_{F_{1}} and (φ_{1})_{F_{1}} are Witt-equivalent. Since dim φ_{1} < (dim ρ_{0})/2, the form φ_{1} is anisotropic over F (ρ_{0}) by Corollary 26.6. As φ_{0} ≺≻ ρ_{0}, the form φ_{1} is also anisotropic over F_{1} = F (φ_{0}) by Corollary 22.17. Therefore, (φ_{F_{1}})_{an} ≃ (φ_{1})_{F_{1}}. This proves the last statement for i = 1. Both statements of the proposition follow now by induction on r. □

The notion of an excellent form was introduced by Knebusch.

29. Excellent field extensions

A field extension E/F is called excellent if the anisotropic part φ_{E} of any quadratic form φ over F is defined over F, i.e., there is a quadratic form ψ over F satisfying (φ_{E})_{an} ≃ ψ_{E}. The concept of an excellent field extension was introduced in [39] and further information about this concept can be found there.

Example 29.1. Suppose that every anisotropic form over F remains anisotropic over E. Then for every quadratic form φ over F the form (φ_{an})_{E} is anisotropic and therefore is isometric to the anisotropic part of φ_{E}. It follows that E/F is an excellent field extension. In particular, it follows from Lemma 7.15 and Springer’s Theorem 18.5 that purely transcendental field extensions and odd degree field extensions are excellent.

Example 29.2. Let E/F be a separable quadratic field extension. Then E = F (σ), where σ is the (nondegenerate) binary norm form of E/F. It follows from Corollary 22.12 that E/F is an excellent field extension.

Example 29.3. Let E/F be a field extension such that every quadratic form over E is defined over F. Then E/F is obviously an excellent extension.

Exercise 29.4. Let E be either an algebraic closure or a separable closure of a field F. Prove that every quadratic form over E is defined over F. In particular, E/F is an excellent extension.

Let ρ be an irreducible nondegenerate quadratic form over F. If dim ρ = 2, the extension F (ρ)/F is separable quadratic and therefore is excellent by Example 29.2. We extend this result to nondegenerate forms of dimension 3.

Notation 29.5. Until the end of this section, let K/F be a separable quadratic field extension and let a ∈ F^{×}. Consider the 3-dimensional quadratic form ρ = N_{K/F} ⊥ ⟨−a⟩ on the space U := K ⊕ F. Let X be the projective quadric of ρ. It is a smooth conic curve in ℙ(U). In the projective coordinates [s : t] on K ⊕ F, the conic X is given by the equation N_{K/F} (s) = at^{2}. We write E for the field F (ρ) = F (X).

The intersection of X with ℙ(K) is Spec F (x) for a point x ∈ X of degree 2 with F (x) ≃ K. In fact, Spec F (x) is the quadric of the form N_{K/F} = ρ|_{K}. Over K the norm form N_{K/F} (s) factors into a product s · s′ of linear forms. Therefore, there are two rational points y and y′ of the curve X_{K} mapping to x under the natural morphism X_{K} → X, so that the divisor div(s/t) equals y − y′ and div(s′/t) equals y′ − y. Moreover, we have

(29.6)  N_{KE/E}(s/t) = N_{K/F} (s)/t^{2} = at^{2}/t^{2} = a.

For any n ≥ 0, let L_{n} be the F -subspace

{f ∈ E^{×} | div(f) + nx ≥ 0} ∪ {0}

of E. We have

F = L_{0} ⊂ L_{1} ⊂ L_{2} ⊂ ⋯ ⊂ E

and L_{n} · L_{m} ⊂ L_{n+m} for all n, m ≥ 0. In particular, the union L of all L_{n} is a subring of E. In fact, E is the quotient field of L.

In addition, O_{X,x} · L_{n} ⊂ L_{n} and 𝔪_{X,x} · L_{n} ⊂ L_{n−1} for every n ≥ 1 where O_{X,x} is the local ring of x and 𝔪_{X,x} its maximal ideal. In particular, we have the structure of a K-vector space on L_{n}/L_{n−1} for every n ≥ 1.

Set \overline{L}_{n} = L_{n}/L_{n−1} for n ≥ 1 and \overline{L}_{0} = K. The graded group \overline{L}_{∗} has the structure of a K-algebra.

The following lemma is an easy case of the Riemann-Roch Theorem.

Lemma 29.7. In the notation above, we have dim_{K}(\overline{L}_{n}) = 1 for all n ≥ 0. Moreover, \overline{L}_{∗} is a polynomial ring over K in one variable.

Proof. Let f, g ∈ L_{n} \ L_{n−1} for n ≥ 1. Since f = (f/g)g and f/g ∈ (O_{X,x})^{×}, the images of f and g in \overline{L}_{n} are linearly dependent over K. Hence dim_{K}(\overline{L}_{n}) ≤ 1. On the other hand, for a nonzero linear form l on K, we have div(l/t) = z − x for some z ≠ x. Hence (l/t)^{n} ∈ L_{n} \ L_{n−1} and therefore dim_{K}(\overline{L}_{n}) ≥ 1. Moreover, \overline{L}_{∗} = K[l/t]. □

Proposition 29.8. Let φ : V → F be an anisotropic quadratic form and suppose that for some n ≥ 1 there exists

v ∈ (V ⊗ L_{n}) \ (V ⊗ L_{n−1})

such that φ(v) = 0. Then there exists a subspace W ⊂ V of dimension 2 satisfying:

(1) φ|_{W} is similar to N_{K/F}.

(2) There exists a nonzero ṽ ∈ V ⊗ L_{n−1} such that φ̃(ṽ) = 0 where φ̃ is the quadratic form a(φ|_{W}) ⊥ φ|_{W^{⊥}} on V.

Proof. Let v̄ denote the image of v under the canonical map V ⊗L_{n} → V ⊗\overline{L}_{n}. We have v̄ ≠ 0 as v ∉ V ⊗ L_{n−1}. Since \overline{L}_{n} is 2-dimensional over F by Lemma 29.7, there is a subspace W ⊂ V of dimension 2 with v̄ ∈ W ⊗ \overline{L}_{n}.

As v̄ is an isotropic vector in W ⊗ \overline{L}_{∗} and \overline{L}_{∗} is a polynomial algebra over K, we have W ⊗ K is isotropic. It follows from Corollary 22.12 that the restriction φ|_{W} is isometric to c N_{K/F} for some c ∈ F^{×} and, in particular, nondegenerate.

By Proposition 7.22, we can write v = w+w′ with w ∈ W ⊗L_{n} and w′ ∈ W^{⊥} ⊗ L_{n}. By construction of W, we have w̄′ = 0 in V ⊗\overline{L}_{n}, i.e., w′ ∈ V ⊗L_{n−1}, therefore φ(w′) ∈ L_{2n−2}. Since 0 = φ(v) = φ(w) + φ(w′), we must have φ(w) ∈ L_{2n−2}.

We may therefore assume that W = K and φ|_{K} = c N_{K/F}. Thus we have w ∈ K ⊗ L_{n} ⊂ K ⊗ E = K(X). Considering w as a function on X_{K} we have div_{∞}(w) = my + m′y′ for some m, m′ ≤ n where div_{∞} is the divisor of poles. As w ∉ W ⊗ L_{n−1}, we must have one of the numbers m and m′, say m, equal to n.

Let σ be the generator of the Galois group of K/F. We have σ(y) = y′, hence div_{∞}(σw) = my′ + m′y and

div_{∞}(φ(w)) = div_{∞}(N_{K/F} (w)) = div_{∞}(w) + div_{∞}(σw) = (m + m′)(y + y′).

As φ(w) ∈ L_{2n−2}, we have m + m′ ≤ 2n − 2, i.e., m′ ≤ n − 2.

Note also that

div_{∞}(ws/t) = div_{∞}(w) + y − y′ = (m − 1)y + (m′ + 1)y′.

As both m − 1 and m′ + 1 are at most n − 1, we have ws/t ∈ K ⊗ L_{n−1}.

Now let φ̃ be the quadratic form a(φ|_{W}) ⊥ φ|_{W ⊥} on V = W ⊕ W^{⊥} and set ṽ = a^{−1}ws/t + w′ ∈ V ⊗ L_{n−1}. By (29.6) we have

φ̃(ṽ) = aφ(a^{−1}ws/t) + φ(w′)
= a^{−1} N_{K(X)/F (X)}(s/t)φ(w) + φ(w′) = φ(w) + φ(w′) = 0.  □

Corollary 29.9. Let φ be a quadratic form over F such that φ_{E} is isotropic. Then there exists an isotropic quadratic form ψ over F such that ψ_{E} ≃ φ_{E}.

Proof. Let v ∈ V ⊗ E be an isotropic vector of φ_{E}. Scaling v we may assume that v ∈ V ⊗ L. Choose the smallest n such that v ∈ V ⊗ L_{n}. We induct on n. If n = 0, i.e., v ∈ V, the form φ is isotropic and we can take ψ = φ.

Suppose that n ≥ 1. By Proposition 29.8, there exist a 2-dimensional subspace W ⊂ V such that φ|_{W} is similar to N_{K/F} and an isotropic vector ṽ ∈ V ⊗ L_{n−1} for the quadratic form φ̃ = a(φ|_{W}) ⊥ (φ|_{W^{⊥}}) on V. As a is the norm in the quadratic extension KE/E, the forms N_{K/F} and aN_{K/F} are isometric over E, hence φ̃_{E} ≃ φ_{E}. By the induction hypothesis applied to the form φ̃, there is an isotropic quadratic form ψ over F such that ψ_{E} ≃ φ̃_{E} ≃ φ_{E}. □

Theorem 29.10. Let ρ be a nondegenerate 3-dimensional quadratic form over F. Then the field extension F (ρ)/F is excellent.

Proof. We may assume ρ is the form in Notation 29.5 as every nondegenerate 3-dimensional quadratic form over F is similar to such a form. Let E = F (ρ) and let φ be a quadratic form over F. We show by induction on dim φ_{an} that (φ_{E})_{an} is defined over F. If φ_{an} is anisotropic over E we are done as (φ_{E})_{an} ≃ (φ_{an})_{E}.

Suppose that φ_{an} is isotropic over E. By Corollary 29.9 applied to φ_{an}, there exists an isotropic quadratic form ψ over F such that ψ_{E} ≃ (φ_{an})_{E}. As dim ψ_{an} < dim φ_{an}, by the induction hypothesis, there is a quadratic form μ over F such that (ψ_{E})_{an} ≃ μ_{E}. Since μ_{E} ∼ ψ_{E} ∼ φ_{E}, we have (φ_{E})_{an} ≃ μ_{E}. □

Corollary 29.11. Let φ ∈ GP_{2}(F). Then F (φ)/F is excellent.

Proof. Let ψ be a Pfister neighbor of φ of dimension three. Let K = F (φ) and L = F (ψ). By Remark 23.11 and Proposition 22.9, the field extensions KL/K and KL/L are purely transcendental. Let ν be a quadratic form over F. By Theorem 29.10, there exists a quadratic form σ over F such that (ν_{L})_{an} ≃ σ_{L}. Hence

((ν_{K})_{an})_{KL} ≃ (ν_{KL})_{an} ≃ ((ν_{L})_{an})_{KL} ≃ σ_{KL}.

It follows that (ν_{K})_{an} ≃ σ_{K}. □

The corollary above was first proven by Arason (cf. [39, Appendix II]). We have followed here Rost’s approach in [116]. This result does not generalize to higher fold Pfister forms. It is known, in general, for every n > 2, there exists a field F and a φ ∈ GP_{n}(F) with F (φ)/F not an excellent extension (cf. [61]).

30. Central simple algebras over function fields of quadratic forms

Let D be a finite dimensional division algebra over a field F. Let D[t] denote the F [t]-algebra D ⊗_{F} F [t] and D(t) denote the F (t)-algebra D ⊗_{F} F (t). As D(t) has no zero divisors and is finite dimensional over F (t), it is a division algebra. The main result in this section is Theorem 30.5. This was originally proved in [102] for fields of characteristic different from 2 using Swan’s calculation of Grothendieck group of a smooth projective quadric. We generalize the elementary proof for this case given by Tignol in [131].

A subring A ⊂ D(t) is called an order over F [t] if it is a finitely generated F [t]-submodule of D(t).

Lemma 30.1. Let D be a finite dimensional division F -algebra. Then every order A ⊂ D(t) over F [t] is conjugate to a subring of D[t].

Proof. As A is finitely generated as an F [t]-module, there is a nonzero f ∈ F [t] such that Af ⊂ D[t]. The subset DAf of D[t] is a left ideal. The ring D[t] admits both the left and the right Euclidean algorithm relative to degree. In follows that all one-sided ideals in D[t] are principal. In particular, DAf = D[t]x for some x ∈ D[t]. As A is a ring, for every y ∈ A, we have

xy ∈ D[t]xy = DAfy ⊂ DAf = D[t]x,

hence xyx^{−1} ∈ D[t]. Thus xAx^{−1} ⊂ D[t]. □

Lemma 30.2. Suppose that R is a commutative ring and S a (not necessarily commutative) R-algebra. Let X ⊂ S be an R-submodule generated by n elements. Suppose that every x ∈ X satisfies the equation x^{2} + ax + b = 0 for some a, b ∈ R. Then the R-subalgebra of S generated by X can be generated as an R-module by 2^{n} elements.

Proof. Let x_{1}, …, x_{n} be generators for the R-module X. Writing quadratic equations for every pair of generators x_{i}, x_{j} and x_{i} + x_{j}, we see that x_{i}x_{j} + x_{j}x_{i} + ax_{i} + bx_{j} + c = 0 for some a, b, c ∈ R. Therefore, the R-subalgebra of S generated by X is generated as an R-module by the monomials x_{i_{1}}x_{i_{2}} … x_{i_{k}} with i_{1} < i_{2} < ⋯ < i_{k}. □

Let φ be a quadratic form on V over F and v_{0} ∈ V a vector such that φ(v_{0}) = 1. For every v ∈ V, the element −vv_{0} in the even Clifford algebra C_{0}(φ) satisfies the quadratic equation

(30.3)  (−vv_{0})^{2} + b_{φ}(v_{0}, v)(−vv_{0}) + φ(v) = 0.

Choose a subspace U ⊂ V such that V = F v_{0} ⊕ U. Let J be the ideal of the tensor algebra T (U) generated by the elements v ⊗ v + b_{φ}(v_{0}, v)v + φ(v) for all v ∈ U.

Lemma 30.4. With U as above, the F -algebra homomorphism α : T (U)/J → C_{0}(φ) defined by α(v + J) = −vv_{0} is an isomorphism.

Proof. By Lemma 30.2, we have dim T (U)/J ≤ 2^{dim U} = dim C_{0}(φ). As α is surjective, it must be an isomorphism. □

Theorem 30.5. Let D be a finite dimensional division F -algebra and φ an irreducible quadratic form over F. Then D_{F (φ)} is not a division algebra if and only if there is an F -algebra homomorphism C_{0}(φ) → D.

Proof. Scaling φ, we may assume that there is v_{0} ∈ V satisfying φ(v_{0}) = 1 where V = V_{φ}. We use the decomposition V = F v_{0} ⊕ U as above and set

l(v) = b_{φ}(v_{0}, v) for every v ∈ U.

Claim: Suppose that D_{F (φ)} is not a division algebra. Then there is an F -linear map f : U → D satisfying the equality of quadratic maps

(30.6)  f^{2} + lf + φ = 0.

(We view the left hand side as the quadratic map v ↦ f(v)^{2} + l(v)f(v) + φ(v) on U). If we establish the claim, then the map f extends to an F -algebra homomorphism T (U)/J → D and, by Lemma 30.4, we get an F -algebra homomorphism C_{0}(φ) → D as needed.

We prove the claim by induction on dim U. Suppose that dim U = 1, i.e., U = F v for some v. By Example 22.2, we have F (φ) ≃ C_{0}(φ) = F ⊕ F x with x satisfying the quadratic equation x^{2} + ax + b = 0 with a = l(v) and b = φ(v) by equation (30.3). Since D_{F (φ)} is not a division algebra, there exists a nonzero element d′ + dx ∈ D_{F (φ)} with d, d′ ∈ D such that (d′ + dx)^{2} = 0 or equivalently d′^{2} = bd^{2} and dd′ + d′d = ad^{2}. Since D is a division algebra, we have d ≠ 0. Then the element d′d^{−1} in D satisfies

(d′d^{−1})^{2} − a(d′d^{−1}) + b = 0.

Therefore, the assignment v ↦ −d′d^{−1} gives rise to the desired map f : U → D. Now consider the general case dim U ≥ 2. Choose a decomposition

U = F v_{1} ⊕ F v_{2} ⊕ W

for some nonzero v_{1}, v_{2} ∈ U and a subspace W ⊂ U and set V′ = F v_{0} ⊕ F v_{1} ⊕ W, U′ = F v_{1} ⊕W so that V′ = F v_{0} ⊕U′. Consider the quadratic form φ′ on the vector space V′_{F (t)} over the function field F (t) defined by

φ′(av_{0} + bv_{1} + w) = φ(av_{0} + bv_{1} + btv_{2} + w).

We show that the function fields F (φ) and F (t)(φ′) are isomorphic over F. Indeed, consider the injective F -linear map θ : V^{∗} → V′_{F (t)}^{∗} taking a linear functional z to the functional z′ defined by z′(av_{0} + bv_{1} + w) = z(av_{0} + bv_{1} + btv_{2} + w). The map θ identifies the ring S^{•}(V^{∗}) with a graded subring of S^{•}(V′_{F (t)}^{∗}) so that φ is identified with φ′. Let x_{1} and x_{2} be the coordinate functions of v_{1} and v_{2} in V, respectively, and x′_{1} the coordinate function of v_{1} in V′. We have x_{1} = x′_{1} and x_{2} = tx′_{1} in S^{1}(V′_{F (t)}^{∗}). Therefore, the localization of the ring S^{•}(V^{∗}) with respect to the multiplicative system F [x_{1}, x_{2}] \ {0} coincides with the localization of S^{•}(V′_{F (t)}^{∗}) with respect to F (t)[x′_{1}] \ {0}. Note that F [x_{1}, x_{2}] ∩ (φ) = 0 and F (t)[x′_{1}] ∩ (φ′) = 0. It follows that the localizations S^{•}(V^{∗})_{(φ)} and S^{•}(V′_{F (t)}^{∗})_{(φ′)} are equal. As the function fields F (φ) and F (t)(φ′) coincide with the degree 0 components of the quotient fields of their respective localizations, the assertion follows.

Let l′(v) = b′_{φ}(v_{0}, v), so

l′(av_{0} + bv_{1} + w) = l(av_{0} + bv_{1} + btv_{2} + w).

Applying the induction hypothesis to the quadratic form φ′ over F (t) and the F (t)-algebra D_{F (t)} produces an F (t)-linear map f′ : U′_{F (t)} → D_{F (t)} satisfying

(30.7)  f′^{2} + l′f′ + φ′ = 0.

Consider the F [t]-submodule X = f′(U′_{F [t]}) in D_{F [t]}. By Lemma 30.2, the F [t]- subalgebra generated by X is a finitely generated F [t]-module. It follows from Lemma 30.1 that, after applying an inner automorphism of D_{F (t)}, we have f′(v) ∈ D_{F [t]} for all v. Considering the highest degree terms of f′ (with respect to t) and taking into account the fact that D is a division algebra, we see that deg f′ ≤ 1, i.e., f′ = g + ht for two linear maps g, h : U′ → D. Comparison of degree 2 terms of (30.7) yields

h(v)^{2} + bl(v_{2})h(v) + b^{2}φ(v_{2}) = 0

for all v = bv_{1} + w. In particular, h is zero on W, therefore h(v) = bh(v_{1}). Thus (30.7) reads

(30.8)  (g(v) + bth(v_{1}))^{2} + l(bv_{1} + btv_{2})(g(v) + bth(v_{1})) + φ(v + btv_{2}) = 0

for every v = bv_{1} + w. Let f : U → D be the F -linear map defined by the formula

f(bv_{1} + cv_{2} + w) = g(bv_{1} + w) + ch(v_{1}).

Substituting c/b for t in (30.8), we see that (30.6) holds on all vectors bv_{1} + cv_{2} + w with b ≠ 0 and therefore holds as an equality of quadratic maps. The claim is proven.

We now prove the converse. Suppose that there is an F -algebra homomorphism s : C_{0}(φ) → D. Consider the two F -linear maps p, q : V → D given by p(v) = s(vv_{0}) and q(v) = s(vv_{0} − l(v)). We have

p(v)q(v) = s((vv_{0})^{2} − l(v)vv_{0}) = s(φ(v)) = φ(v)

by equation (30.3). It follows that p and q are injective maps if φ is anisotropic. The maps p and q stay injective over any field extension. Let L/F be a field extension such that φ_{L} is isotropic (e.g., L = F (φ)). Then if v′ ∈ V_{L} is a nonzero isotropic vector, we have p(v′)q(v′) = φ(v′) = 0, but p(v′) ≠ 0 and q(v′) ≠ 0. It follows that D_{L} is not a division algebra.

It remains to consider the case when φ is isotropic. We first show that every isotropic vector v ∈ V belongs to rad b_{φ}. Suppose this is not true. Then there is a u ∈ V satisfying b_{φ}(v, u) ≠ 0. Let H be the 2-dimensional subspace generated by v and u. The restriction of φ on H is a hyperbolic plane. Let w ∈ V be a nonzero vector orthogonal to H and let a = φ(w). Then

𝕄_{2}(F) = C(−aH) = C_{0}(F w ⊥ H) ⊂ C_{0}(φ)

by Proposition 11.4. The image of the matrix algebra 𝕄_{2}(F) under s is isomorphic to 𝕄_{2}(F) and therefore contains zero divisors, a contradiction proving the assertion.

Let V′ be a subspace of V satisfying V = rad φ ⊕ V′. As every isotropic vector belongs to rad b_{φ}, the restriction φ′ of φ on V′ is anisotropic. The natural map C_{0}(φ) → C_{0}(φ′) induces an isomorphism C_{0}(φ)/J →^{∼} C_{0}(φ′), where J = (rad φ)C_{1}(φ). Since x^{2} = 0 for every x ∈ rad φ, we have s(J) = 0. Therefore, s induces an F -algebra homomorphism s′ : C_{0}(φ′) → D. By the anisotropic case, D is not a division algebra over F (φ′). Since F (φ) is a field extension of F (φ′), the algebra D_{F (φ)} is also not a division algebra. □

Corollary 30.9. Let D be a division F -algebra of dimension less than 2^{2n} and φ a nondegenerate quadratic form of dimension at least 2n + 1 over F. Then D_{F (φ)} is also a division algebra.

Proof. Let ψ be a nondegenerate subform of φ of dimension 2n + 1. As F (ψ)(φ)/F (ψ) is a purely transcendental extension by Proposition 22.9, we may replace φ by ψ and assume that dim φ = 2n + 1. By Proposition 11.6, the algebra C_{0}(φ) is simple of dimension 2^{2n}. If D_{F (φ)} is not a division algebra, then there is an F -algebra homomorphism C_{0}(φ) → D by Theorem 30.5. This homomorphism must be injective as C_{0}(φ) is simple. But this is impossible by dimension count. □

Corollary 30.10. Let D be a division F -algebra and let φ be a nondegenerate quadratic form over F satisfying:

(1) If dim φ is odd or φ ∈ I_{q}(F) \ I_{q}^{2}(F), then C_{0}(φ) is not a division algebra.

(2) If φ ∈ I_{q}^{2}(F), then C^{+}(φ) is not a division algebra over F (cf. Remark

13.9).

Then D_{F (φ)} is a division algebra.

Proof. If D_{F (φ)} is not a division algebra, there is an F -algebra homomorphism f : C_{0}(φ) → D by Theorem 30.5. If φ ∈ I_{q}^{2}(F) we have C_{0}(φ) ≃ C^{+}(φ) × C^{+}(φ) by Remark 13.9. Thus in every case the image of f lies in a nondivision subalgebra of D. Therefore, D is not a division algebra, a contradiction. □

Corollary 30.11. Let D be a division F -algebra and let φ ∈ I_{q}^{3}(F) be a nonzero quadratic form. Then D_{F (φ)} is a division algebra.

Proof. By Theorem 14.3, the Clifford algebra C(φ) is split. In particular, C^{+}(φ) is not a division algebra. The statement follows now from Corollary 30.10.

□

CHAPTER V Bilinear and Quadratic Forms and Algebraic Extensions

31. Structure of the Witt ring

In this section, we investigate the structure of the Witt ring of nondegenerate symmetric bilinear forms. For fields F whose level s(F) is finite, i.e., nonformally real fields, the ring structure is quite simple. The Witt ring of such a field has a unique prime ideal, viz., the fundamental ideal and W (F) (as an abelian group) has exponent 2s(F). As s(F) = 2^{n} for some nonnegative integer this means that the Witt ring is 2-primary torsion. The case of formally real fields F, i.e., fields of infinite level, is more involved. Orderings on such a field give rise to prime ideals in W (F). The torsion in W (F) is still 2-primary, but this is not as easy to show. Therefore, we do the two cases separately.

31.A. Nonformally real fields. We consider the case of nonformally real fields first.

A field F is called quadratically closed if F = F^{2}. For example, algebraically closed fields are quadratically closed. A field of characteristic 2 is quadratically closed if and only if it is perfect. Over a quadratically closed field, the structure of the Witt ring is very simple. Indeed, we have

Lemma 31.1. For a field F the following are equivalent:

(1) F is quadratically closed.

(2) W (F) = ℤ/2ℤ.

(3) I(F) = 0.

Proof. As W (F)/I(F) = ℤ/2ℤ, we have W (F) ≃ ℤ/2ℤ if and only if I(F) = 0 if and only if ⟨1, −a⟩ = 0 in W (F) for all a ∈ F^{×} if and only if a ∈ F^{×2} for all a ∈ F^{×}. □

Example 31.2. Let F be a finite field with char F = p and |F | = q. If p = 2, then F = F^{2} and F is quadratically closed. So suppose that p > 2. Then F^{×2} ≃ F^{×}/\{±1\} so |F^{2}| = \frac{1}{2}(q + 1). Let F^{×}/F^{×2} = \{F^{×2}, aF^{×2}\}. Since the finite sets F^{2} and \{a − y^{2} | y ∈ F \} both have \frac{1}{2}(q + 1) elements, they intersect nontrivially. It follows that every element in F is a sum of two squares. We have −1 ∈ F^{×2} if and only if q ≡ 1 mod 4.

If q ≡ 3 mod 4, then −1 ∉ F^{×2} and the level s(F) of F is 2. We may assume that a = −1. Then ⟨1, 1, 1⟩ = ⟨1, −1, −1⟩ = ⟨−1⟩ in W (F) so W (F) equals {0, ⟨1⟩, ⟨−1⟩, ⟨1, 1⟩} and is isomorphic to the ring ℤ/4ℤ.

If q ≡ 1 mod 4, then −1 is a square and W (F) equals {0, ⟨1⟩, ⟨a⟩, ⟨1, a⟩} and is isomorphic to the group ring ℤ/2ℤ[F^{×}/F^{×2}].

It follows from Example 31.2 that s(F) = 1 or 2 for any field F of positive characteristic. In general, if F is not formally real, s(F) = 2^{n} by Corollary 6.8. There exist fields of level 2^{m} for each m ≥ 1.

Lemma 31.3. Let 2^{m} ≤ n < 2^{m+1}. Suppose that F satisfies s(F) > 2^{m}, e.g., F is formally real, and φ = (n + 1)⟨1⟩_{q}. Then s(F (φ)) = 2^{m}.

Proof. As s(F) > 1, the characteristic of F is not 2. Since φ_{F (φ)} is isotropic, it follows that s(F (φ)) ≤ 2^{m} by Corollary 6.8. If φ was isotropic over F, then s(F) = s(F (φ)) ≤ 2^{m} as F (φ)/F is purely transcendental by Proposition 22.9. This contradicts the hypothesis. So φ is anisotropic. If s(F (φ)) < 2^{m}, then (2^{m}⟨1⟩)_{F (φ)} is a Pfister form as char F ≠ 2, hence is hyperbolic. It follows that 2^{m} = dim 2^{m}⟨1⟩ ≥ dim φ > 2^{m} by the Subform Theorem 22.5, a contradiction. □

The ring structure of W (F) due to Pfister (cf. [110]) is given by the following:

Proposition 31.4. Let F be nonformally real with s(F) = 2^{n}. Then:

(1) Spec(W (F)) = {I(F)}.

(2) W (F) is a local ring of Krull dimension zero with maximal ideal I(F).

(3) nil(W (F)) = rad(W (F)) = zd(F) = I(F).

(4) W (F)^{×} = {𝔟 | dim 𝔟 is odd}.

(5) W (F) is connected, i.e., 0 and 1 are the only idempotents in W (F).

(6) W (F) is a 2-primary torsion group of exponent 2s(F).

(7) W (F) is artinian if and only if it is noetherian if and only if |F^{×}/F^{×2}| is finite if and only if W (F) is a finite ring.

Proof. Let s = s(F). The integer 2s is the smallest integer such that the bilinear Pfister form 2s⟨1⟩_{b} is metabolic, hence zero in the Witt ring. Therefore, 2^{n+1}⟨a⟩ = 0 in W (F) for every a ∈ F^{×}. It follows that W (F) is 2-primary torsion of exponent 2^{n+1}, i.e., (6) holds. As

⟨⟨a⟩⟩^{n+2} = ⟨⟨a, …, a⟩⟩ = ⟨⟨a, −1, …, −1⟩⟩ = 2^{n+1}⟨⟨a⟩⟩ = 0

in W (F) for every a ∈ F^{×} by Example 4.16, we have I(F) lies in every prime ideal. Since W (F)/I(F) ≃ ℤ/2ℤ, the fundamental ideal I(F) is maximal, hence is the only prime ideal. This proves (1). As I(F) is the only prime ideal (2) − (5) follows easily.

Finally, we show (7). Suppose that W (F) is noetherian. Then I(F) is a finitely generated W (F)-module so I(F)/I^{2}(F) is a finitely generated W (F)/I(F)- module. As F^{×}/F^{×2} ≃ I(F)/I^{2}(F) by Proposition 4.13 and ℤ/2ℤ ≃ W (F)/I(F), we have F^{×}/F^{×2} is finite. Conversely, suppose that F^{×}/F^{×2} is finite. By (2.6), we have a ring epimorphism ℤ[F^{×}/F^{×2}] → W (F). As the group ring ℤ[F^{×}/F^{×2}] is noetherian, W (F) is noetherian. As 2sW (F) = 0 and W (F) is generated by the classes of 1-dimensional forms, we see that |W (F)| ≤ |F^{×}/F^{×2}|^{2s}. Statement (7) now follows easily. □

We turn to formally real fields, i.e., those fields of infinite level. In particular, formally real fields are of characteristic 0, so the theories of symmetric bilinear forms and quadratic forms merge. The structure of the Witt ring of a formally real field is more complicated as well as more interesting. We shall use the basic algebraic and topological structure of formally real fields which can be found in Appendices §95 and §96. Recall that a formally field F is called euclidean if every element in F^{×} is a square or minus a square. So F is euclidean if and only if F is formally real and F^{×}/F^{×2} = {F^{×2}, −F^{×2}}. In particular, every real-closed field (cf. §95) is euclidean. Sylvester’s Law of Inertia for real-closed fields generalizes to euclidean fields.

Proposition 31.5 (Sylvester’s Law of Inertia). Let F be a field. Then the following are equivalent:

(1) F is euclidean.

(2) F is formally real and if 𝔟 is a nondegenerate symmetric bilinear form, there exists unique nonnegative integers m, n such that 𝔟 ≃ m⟨1⟩ ⊥ n⟨−1⟩.

(3) W (F) ≃ ℤ as rings.

(4) F^{2} is an ordering of F.

Proof. (1) ⇒ (2): As F is formally real, char F = 0 so every bilinear form is diagonalizable. Since F^{×}/F^{×2} = {F^{×2}, −F^{×2}}, every nondegenerate bilinear form is isometric to m⟨1⟩ ⊥ n⟨−1⟩ for some nonnegative integers n and m. The integers n and m are unique by Witt Cancellation 1.28.

(2) ⇒ (3): By (2) every anisotropic quadratic form is isometric to r⟨1⟩ for some unique integer r. As F is formally real every such form is anisotropic.

(3) ⇒ (4): Let sgn : W (F) → ℤ be the isomorphism. Then sgn⟨1⟩ = 1 so ⟨1⟩ has infinite order, hence F is formally real. Let a ∈ F. Then sgn⟨a⟩ = n for some integer n. Thus ⟨a⟩ = n⟨1⟩ in W (F). In particular, n is odd. Taking determinants, we must have aF^{×2} = ±F^{×2}. It follows that F^{×}/F^{×2} = {F^{×2}, −F^{×2}}. As F is formally real, F^{2} + F^{2} ⊂ F^{2}, hence F^{2} is an ordering.

(4) ⇒ (1): As F has an ordering, it is formally real. As F^{2} is an ordering, F = F^{2} ∪ (−F^{2}) with −1 ∉ F^{2}, so F is euclidean. □

31.B. Pythagorean fields. We turn to a class of fields important in the study of formally real fields that generalize quadratically closed fields.

Let F be a euclidean field. If 𝔟 is a nondegenerate symmetric bilinear form, then 𝔟 ≃ m⟨1⟩ ⊥ n⟨−1⟩ for unique nonnegative integers n and m. The integer m − n is called the signature of 𝔟 and denoted sgn 𝔟. This induces an isomorphism sgn : W (F) → ℤ taking the Witt class of 𝔟 to sgn 𝔟 called the signature map.

Let F be a field. Set

D(∞⟨1⟩) := ⋃_{n} D(n⟨1⟩) = {x | x is a nonzero sum of squares in F },
D̃(∞⟨1⟩) := D(∞⟨1⟩) ∪ {0}.

A field F of characteristic different from 2 is called a pythagorean field if every sum of squares of elements in F is itself a square, i.e., D̃(∞⟨1⟩) = F^{2}. A field F of characteristic 2 is called pythagorean if F is quadratically closed, i.e., perfect.

Example 31.6. Let F be a field.

(1) Every euclidean field is pythagorean.

(2) Let F be a field of characteristic different from 2 and K = F ((t)), a Laurent series field over F. Then K is the quotient field of F [[t]], a complete discrete valuation ring. If F is formally real, then so is K as n⟨1⟩ is anisotropic over K for all n by Lemma 19.4. Suppose that F is formally real and pythagorean. If x_{i} ∈ K^{×} for i = 1, 2, then there exist integers m_{i} such that x_{i} = t^{m_{i}}(a_{i} + ty_{i}) with a_{i} ∈ F^{×} and y_{i} ∈ F [[t]] for i = 1, 2. Suppose that m_{1} ≤ m_{2}, then x_{1}^{2} + x_{2}^{2} = t^{2m_{1}}(c + tz) with z ∈ F [[t]] and c = a_{1}^{2} if m_{1} < m_{2} and c = a_{1}^{2} + a_{2}^{2} if m_{1} = m_{2}, hence c is a square in F in either case. As K is formally real, c ≠ 0 in either case. Hence c + tz is a square in K by Hensel’s Lemma. It follows that K is also pythagorean. In particular, the finitely iterated Laurent series field F_{n} = F ((t_{1})) ⋯ ((t_{n})) as well as the infinitely iterated Laurent series field F_{∞} = colim F_{n} = F ((t_{1})) ⋯ ((t_{n})) ⋯ are formally real and pythagorean if F is.

(3) If F is not formally real and char F ≠ 2, then s = s(F) is finite so the symmetric bilinear form (s + 1)⟨1⟩ is isotropic, hence universal by Corollary 1.25. Therefore, F = D̃(∞⟨1⟩). It follows that if F is not formally real, then F is pythagorean if and only if it is quadratically closed. In particular, if F is not formally real, then the field F ((t)) is not pythagorean as t is not a square.

Exercise 31.7. Let F be a formally real pythagorean field and let 𝔟 be a bilinear form over F. Prove that the set D(𝔟) is closed under addition.

Proposition 31.8. Let F be a field. Then the following are equivalent: (1) F is pythagorean.

(2) I(F) is torsion-free.

(3) There are no anisotropic torsion binary bilinear forms over F.

Proof. (1) ⇒ (2): If s(F) is finite, then F is quadratically closed so W (F) = {0, ⟨1⟩} and I(F) = 0. Therefore, we may assume that F is formally real. We show in this case that W (F) is torsion-free. Let 𝔟 be an anisotropic bilinear form over F that is torsion in W (F), say m𝔟 = 0 in W (F) for some positive integer m. As 𝔟 is diagonalizable by Corollary 1.19, suppose that 𝔟 ≃ ⟨a_{1}, …, a_{n}⟩ with a_{i} ∈ F^{×}. The form m𝔟_{i} is isotropic so there exists a nontrivial equation ∑_{j} ∑_{i} a_{i}x_{ij}^{2} = 0 in F. As F is pythagorean, there exist x_{i} ∈ F satisfying x_{i}^{2} = ∑_{j} x_{ij}^{2}. Since F is formally real, not all the x_{i} can be zero. Thus (x_{1}, …, x_{n}) is an isotropic vector for 𝔟, a contradiction.

(2) ⇒ (3) is trivial.

(3) ⇒ (1): Let 0 ≠ z ∈ D(2⟨1⟩). Then 2⟨⟨z⟩⟩ = 0 in W (F) by Corollary 6.6. By assumption, ⟨⟨z⟩⟩ = 0 in W (F), hence z ∈ F^{×2}. □

Corollary 31.9. A field F is formally real and pythagorean if and only if W (F) is torsion-free.

Proof. Suppose that W (F) is torsion-free. Then I(F) is torsion-free so F is pythagorean. As ⟨1⟩ is not torsion, s(F) is infinite, hence F is formally real.

Conversely, suppose that F is formally real and pythagorean. Then the proof of (1) ⇒ (2) in Proposition 31.8 shows that W (F) is torsion-free. □

Lemma 31.10. The intersection of pythagorean fields is pythagorean.

Proof. Let F = ⋂_{I} F_{i} with each F_{i} pythagorean. If z = x^{2} +y^{2} with x, y ∈ F, then for each i ∈ I there exist z_{i} ∈ F_{i} with z_{i}^{2} = z. In particular, z_{i} = ±z_{j} for all i, j ∈ I. Thus z_{j} ∈ ⋂_{I} F_{i} = F for every j ∈ I and z = z_{j}^{2}. □

Exercise 31.11. Let K/F be a finite extension. Show that if K is pythagorean so is F. (Hint: If char F ≠ 2 and a = 1 + x^{2} ∈ F \ F^{2}, let z = a + \sqrt{a} ∈ K. Show z ∈ F (\sqrt{a})^{2}, but N_{F (\sqrt{a})/F} (z) ∉ F^{2}.)

Let F be a field and K/F an algebraic extension. We call K a pythagorean closure of F if K is pythagorean, and if F ⊂ E ⫋ K is an intermediate field, then E is not pythagorean. If F̃ is an algebraic closure of F, then the intersection of all pythagorean fields between F and F̃ is pythagorean by the lemma. Clearly, this is a pythagorean closure of F. In particular, a pythagorean closure is unique (after fixing an algebraic closure). We shall denote the pythagorean closure of F by F_{py}. If F is not a formally real field, then F_{py} is just the quadratic closure of F, i.e., a quadratically closed field K algebraic over F such that if F ⊂ E ⫋ K, then E is not quadratically closed. For example, the quadratic closure of the field of rational numbers ℚ is the field of complex constructible numbers.

Exercise 31.12. Let E be a pythagorean closure of a field F. Prove that E/F is an excellent extension. (Hint: In the formally real case use Exercise 31.7 to show that for any quadratic form φ over F the form (φ_{E})_{an} over E takes values in F.)

We next show how to construct the pythagorean closure of a field.

Definition 31.13. Let F be a field with F̃ an algebraic closure. If K/F is a finite extension in F̃, then we say K/F is admissible if there exists a tower

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = K where
F_{i} = F_{i−1}(\sqrt{z_{i−1}}) with z_{i−1} ∈ D(2⟨1⟩_{F_{i−1}}) (31.14)

from F to K.

Remark 31.15. If F is a formally real field and K an admissible extension of F, then K is formally real by Theorem 95.3 in §95.

Lemma 31.16. Suppose that char F ≠ 2. Let L be the union of all admissible extensions over F. Then L = F_{py}. If F is formally real so is F_{py}.

Proof. Let F̃ be a fixed algebraic closure of F. If E and K are admissible extensions of F, then the compositum of EK of E and K is also an admissible extension. It follows that L is a field. If z ∈ L satisfies z = x^{2} + y^{2}, x, y ∈ L, then there exist admissible extensions E and K of F with x ∈ E and y ∈ K. Then EK(\sqrt{z}) is an admissible extension of F, hence \sqrt{z} ∈ EK(\sqrt{z}) ⊂ L. Therefore, L is pythagorean. Let M be pythagorean with F ⊂ M ⊂ F̃. We show L ⊂ M. Let K/F be admissible. Let (31.14) be a tower from F to K. By induction, we may assume that F_{i} ⊂ M. Therefore, z_{i} ∈ M^{2}, hence F_{i+1} ⊂ M. Consequently, K ⊂ M. It follows that L ⊂ M so L = F_{py}. If F is formally real, then so is L by Remark 31.15. □

If F is an arbitrary field, then the quadratic closure of F can also be constructed by taking the union of all square root towers

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = K where F_{i} = F_{i−1}(\sqrt{z_{i−1}}) with z_{i−1} ∈ F_{i−1}^{×}

over F.

Recall that if K/F is a field extension, then W (K/F) is the kernel of the restriction map r_{K/F} : W (F) → W (K).

Lemma 31.17. Let z ∈ D(2⟨1⟩) \ F^{×2}. If K = F (\sqrt{z}) , then

W (K/F) ⊂ ann_{W (F)}(2⟨1⟩).

Proof. It follows from the hypothesis that ⟨⟨z⟩⟩ is anisotropic, hence K/F is a quadratic extension. As z is a sum of squares and not a square, char F ≠ 2. Therefore, by Corollary 23.6, we have W (K/F) = ⟨⟨z⟩⟩W (F). By Corollary 6.6, we have 2⟨⟨z⟩⟩ = 0 in W (F) and the result follows. □

31.C. Formally real fields. Let

W_{t}(F) := {𝔟 ∈ W (F) | there exists a positive integer n such that n𝔟 = 0},

the additive torsion in W (F). It is an ideal in W (F). We have

Theorem 31.18. Let F be a formally real field.

(1) W_{t}(F) is 2-primary, i.e., all torsion elements of W (F) have exponent a power of 2.

(2) W_{t}(F) = W (F_{py}/F).

Proof. As W (F_{py}) is torsion-free by Corollary 31.9, the torsion subgroup W_{t}(F) lies in W (F_{py}/F), so it suffices to show W (F_{py}/F) is a 2-primary torsion group. Let K be an admissible extension of F as in (31.14). Since F_{py} is the union of admissible extensions by Lemma 31.16, it suffices to show W (K/F) is 2-primary torsion. By Lemma 31.17 and induction, it follows that W (K/F) ⊂ ann_{W (F)}(2^{n}⟨1⟩) as needed. □

Lemma 31.19. Let F be a formally real field and 𝔟 ∈ W (F) satisfy 2^{n}𝔟 ≠ 0 in W (F) for any n ≥ 0. Let K/F be an algebraic extension that is maximal with respect to 𝔟_{K} not having order a power of 2 in W (K). Then K is euclidean. In particular, sgn 𝔟_{K} ≠ 0.

Proof. Suppose K is not euclidean. As 2^{n}⟨1⟩ ≠ 0, the field K is formally real. Since K is not euclidean, there exists an x ∈ K^{×} such that x ∉ (K^{×})^{2}∪−(K^{×})^{2}. In particular, both K(\sqrt{x})/K and K(\sqrt{−x})/K are quadratic extensions. By choice of K, there exists a positive integer n such that 𝔠 := 2^{n}𝔟_{K} satisfies 𝔠_{K(\sqrt{x})} and 𝔠_{K(\sqrt{−x})} are metabolic, hence hyperbolic as char F ≠ 2. By Corollary 23.6, there exist forms 𝔠_{1} and 𝔠_{2} over K satisfying 𝔠 ≃ ⟨⟨x⟩⟩ ⊗ 𝔠_{1} ≃ ⟨⟨−x⟩⟩ ⊗ 𝔠_{2}. As −x⟨⟨x⟩⟩ ≃ ⟨⟨x⟩⟩ and x⟨⟨−x⟩⟩ ≃ ⟨⟨−x⟩⟩, we conclude that x𝔠 ≃ 𝔠 ≃ −x𝔠 and hence that 2𝔠 ≃ 𝔠 ⊥ 𝔠 ≃ x𝔠 ⊥ −x𝔠. Thus 2𝔠 = 0 in W (K). This means that 𝔟_{K} is torsion of order 2^{n+1}, a contradiction. □

Proposition 31.20. The following are equivalent:

(1) F can be ordered, i.e., 𝔛(F), the space of orderings of F, is not empty.

(2) F is formally real.

(3) W_{t}(F) ≠ W (F).

(4) W (F) is not a 2-primary torsion group.

(5) There exists an ideal 𝔄 ⊂ W (F) such that W (F)/𝔄 ≃ ℤ.

(6) There exists a prime ideal 𝔓 in W (F) such that char(W (F)/𝔓) ≠ 2. Moreover, if F is formally real, then for any prime ideal 𝔓 in W (F) which satisfies char(W (F)/𝔓) ≠ 2, the set

P_{𝔓} := {x ∈ F^{×} | ⟨⟨x⟩⟩ ∈ 𝔓} ∪ {0}

is an ordering of F.

Proof. (1) ⇒ (2) is clear.

(2) ⇒ (3): By assumption, −1 ∉ D_{F} (n⟨1⟩) for any n > 0, so ⟨1⟩ ∉ W_{t}(F).

(3) ⇒ (4) is trivial.

(4) ⇒ (5): By assumption there exists 𝔟 ∈ W (F) not having order a power of 2. By Lemma 31.19, there exists K/F with K euclidean. In particular, r_{K/F} is surjective. Therefore, 𝔄 = W (K/F) works by Lemma 31.19 and Sylvester’s Law of Inertia 31.5.

(5) ⇒ (6) is trivial.

(6) ⇒ (1): By Proposition 31.4, the field F is formally real. We show that (6) implies the last statement. This will also prove (1). Let 𝔓 in W (F) be a prime ideal satisfying char(W (F)/𝔓) ≠ 2. We must show

(i) P_{𝔓} ∪ (−P_{𝔓}) = F.

(ii) P_{𝔓} + P_{𝔓} ⊂ P_{𝔓}.
(iii) P_{𝔓} · P_{𝔓} ⊂ P_{𝔓}.
(iv) P_{𝔓} ∩ (−P_{𝔓}) = {0}.
(v) −1 ∉ P_{𝔓}.

Suppose that x ≠ 0 and both ±x ∈ P_{𝔓}. Then ⟨⟨−1⟩⟩ = ⟨⟨−x⟩⟩ + ⟨⟨x⟩⟩ lies in 𝔓 so 2⟨1⟩ + 𝔓 = 0 in W (F)/𝔓, a contradiction. This shows (iv) and (v) hold. As ⟨⟨x, −x⟩⟩ = 0 in W (F), either ⟨⟨x⟩⟩ or ⟨⟨−x⟩⟩ lies in 𝔓, so (i) holds. Next let x, y ∈ P_{𝔓}. Then ⟨⟨xy⟩⟩ = ⟨⟨x⟩⟩ + x⟨⟨y⟩⟩ lies in 𝔓 so xy ∈ 𝔓 which is (iii). Finally, we show that (ii) holds, i.e., x + y ∈ P_{𝔓}. We may assume neither x nor y is zero. This implies that z := x + y ≠ 0, otherwise we have the equation ⟨⟨−1⟩⟩ = ⟨1, x, −x, 1⟩ = ⟨1, −x, −y, 1⟩ = ⟨⟨x⟩⟩ + ⟨⟨y⟩⟩ in W (F) which implies that ⟨⟨−1⟩⟩ lies in 𝔓 contradicting (v). Since ⟨−x, −y⟩ ≃ −z⟨⟨−xy⟩⟩ by Corollary 6.6, we have

2⟨−z⟩ = 2⟨−x, −y, zxy⟩ = ⟨−x, −y, zxy, −z, −zxy, zxy⟩

= ⟨⟨x⟩⟩ + ⟨⟨y⟩⟩ − 2⟨1⟩ − z⟨⟨xy⟩⟩

in W (F). As x, y ∈ P_{𝔓} and xy ∈ P_{𝔓} by (iii), it follows that 2⟨⟨z⟩⟩ ∈ 𝔓 as needed. □

The proposition gives another proof of the Artin-Schreier Theorem that every formally real field can be ordered.

In [94], Lewis gave an elegant proof for the structure of the Witt ring, part of which we present in the following exercise.

Exercise 31.21. Let F be an arbitrary field. Show all of the following: (1) (Leung) Define a polynomial p_{n}(t) in F [t] as follows:

p_{n}(t) = \begin{cases}t(t^{2} − 2^{2})(t^{2} − 4^{2}) ⋯ (t^{2} − n^{2}) & if n is even, \\ (t^{2} − 1^{2})(t^{2} − 3^{2})(t^{2} − 5^{2}) ⋯ (t^{2} − n^{2}) & if n is odd. \end{cases}

If 𝔟 is an n-dimensional nondegenerate symmetric bilinear form over F, then p_{n}(𝔟) = 0 in W (F).

(2) W (F) is integral over ℤ if F is formally real and over ℤ/2^{r}ℤ for some r if F is not formally real. In particular, the Krull dimension of W (F) is one if F is formally real and zero if not.

(3) Using Exercise 33.14 below show that W_{t}(F) is 2-primary.

(4) W (F) has no odd-dimensional zero divisors and if W_{t}(F) ≠ 0, then we have zd(W (F)) = I(F). Moreover, W (F) contains no nontrivial idempotents.

(5) If F is formally real, then W_{t}(F) = nil(W (F)).

31.D. The Local-Global Principle. The main result of this subsection is Theorem 31.22 below due to Pfister (cf. [110]).

Let F be a formally real field and 𝔛(F) the space of orderings. Let P ∈ 𝔛(F) and F_{P} be the real closure of F at P (within a fixed algebraic closure). By Sylvester’s Law of Inertia 31.5, the signature map defines an isomorphism sgn : W (F_{P}) → ℤ. In particular, we have a signature map sgn_{P} : W (F) → ℤ given by sgn_{P} = sgn ∘ r_{F_{P} /F}. This is a ring homomorphism satisfying W_{t}(F) ⊂ Ker(r_{F_{P} /F}) = Ker(sgn_{P}). We let

𝔓_{P} = Ker(sgn_{P}) in Spec(W (F)).

Note if F ⊂ K ⊂ F_{P} and 𝔟 is a nondegenerate symmetric bilinear form, then sgn_{P} 𝔟 = sgn_{F_{P}^{2} ∩K} 𝔟_{K}. In particular, if K is euclidean, then sgn_{P} 𝔟 = sgn 𝔟_{K}.

Theorem 31.22 (Local-Global Principle). The sequence

0 → W_{t}(F) → W (F) ⟶^{(r_{F_{P} /F})} ∏_{𝔛(F)} W (F_{P})

is exact.

Proof. We may assume that F is formally real by Proposition 31.4. We saw above that W_{t}(F) ⊂ Ker(sgn_{P}) for every ordering P ∈ 𝔛(F), so the sequence is a zero sequence. Suppose that 𝔟 ∈ W (F) is not torsion of 2-power order. By Lemma 31.19, there exists a euclidean field K/F with 𝔟_{K} not of 2-power order. As K^{2} ∈ 𝔛(K), we have P = K^{2} ∩ F ∈ 𝔛(F). Thus sgn_{P} 𝔟 = sgn 𝔟_{K} ≠ 0. The result follows. □

Corollary 31.23. The map

𝔛(F) → {𝔓 ∈ Spec(W (F)) | W (F)/𝔓 ≃ ℤ} given by P ↦ 𝔓_{P}

is a bijection.

Proof. Let 𝔓 ⊂ W (F) be a prime ideal such that W (F)/𝔓 ≃ ℤ. As in Proposition 31.20, let P_{𝔓} := {x ∈ F^{×} | ⟨⟨x⟩⟩ ∈ 𝔓} ∪ {0} ∈ 𝔛(F).

Claim: 𝔓 ↦ P_{𝔓} is the inverse, i.e., P = P_{𝔓_{P}} and 𝔓 = 𝔓_{P_{𝔓}}.

If P ∈ 𝔛(F), then certainly P ⊂ P_{𝔓_{P}}, so we must have P = P_{𝔓_{P}} as both are orderings. By definition, we see that the composition W (F) → W (F)/𝔓 ⟶^{∼} ℤ maps ⟨x⟩ to sgn_{P_{𝔓}}⟨x⟩. Hence Ker(sgn_{P_{𝔓}}) = 𝔓. □

The spectrum of the Witt ring can now be determined. This result is due to Lorenz-Leicht (cf. [95]) and Harrison (cf. [49]).

Theorem 31.24. Spec(W (F)) consists of:

(1) The fundamental ideal I(F).

(2) 𝔓_{P} with P ∈ 𝔛(F).

(3) 𝔓_{P,p} := 𝔓_{P} + pW (F) = sgn_{P}^{−1}(pℤ), p an odd prime, with P ∈ 𝔛(F).

Moreover, all these ideals are different. The prime ideals in (1) and (3) are the maximal ideals of W (F). If F is formally real, then the ideals in (2) are the minimal primes of W (F) and 𝔓_{P} ⊂ 𝔓_{P,p} ∩ I(F) for all P ∈ 𝔛(F) and for all odd primes p.

Proof. We may assume that F is formally real by Proposition 31.4. Let 𝔓 be a prime ideal in W (F). Let a ∈ F^{×}. As ⟨⟨a, −a⟩⟩ = 0 in W (F) either ⟨⟨a⟩⟩ ∈ 𝔓 or ⟨⟨−a⟩⟩ ∈ 𝔓. In particular, ⟨a⟩ ≡ ±⟨1⟩ mod 𝔓. Hence W (F)/𝔓 is cyclic generated by ⟨1⟩ + 𝔓, so W (F)/𝔓 ≃ ℤ or ℤ/pℤ for p a prime. If x, y ∈ F^{×}, then ⟨x⟩ and ⟨y⟩ are units in W (F), so do not lie in 𝔓. Suppose that W (F)/𝔓 ≃ ℤ/2ℤ. Then we must have ⟨x, y⟩ ∈ 𝔓 for all x, y ∈ F^{×}, hence 𝔓 = I(F). So suppose that W (F)/𝔓 ≄ ℤ/2ℤ. By Proposition 31.20, the set P = P_{𝔓} lies in 𝔛(F). Since W (F)/𝔓_{P} ≃ ℤ, we have 𝔓_{P} ⊂ 𝔓. Hence 𝔓 = 𝔓_{P} or 𝔓 = 𝔓_{P,p} for a suitable odd prime. As each P ∈ 𝔛(F) determines a unique 𝔓_{P} and 𝔓_{P,p} by Corollary 31.20, the result follows. □

Corollary 31.25. If F is formally real, then dim W (F) = 1 and the map 𝔛(F) → Min Spec(W (F)) given by P ↦ Ker(sgn_{P}) is a homeomorphism.

Proof. As ⟨⟨1⟩⟩ does not lie in any minimal prime, for each a ∈ F^{×} either a ∈ 𝔓_{P} or −a ∈ 𝔓_{P} but not both where P ∈ 𝔛(F). The sets H(a) := {P | −a ∈ P } form a subbase for the topology of 𝔛(F) (cf. §96). As a ∈ P for P ∈ 𝔛(F) if and only if ⟨⟨a⟩⟩ ∈ 𝔓_{P} if and only if 𝔓_{P} lies in the basic open set {𝔓 | a ∉ 𝔓 for 𝔓 ∈ Min(Spec W (F)}, the result follows. □

The structure of the Witt ring in the formally real case due to Pfister (cf. [110]) can now be shown.

Proposition 31.26. Let F be formally real. Then:

(1) nil(W (F)) = rad(W (F)) = W_{t}(F).

(2) W (F)^{×} = {𝔟 | sgn_{P} 𝔟 = ±1 for all P ∈ 𝔛(F)}

= {⟨a⟩ + 𝔠 | a ∈ F^{×} and 𝔠 ∈ I^{2}(F) ∩ W_{t}(F)}.

(3) If F is not pythagorean, then zd(W (F)) = I(F).

(4) If F is pythagorean, then zd(W (F)) = ⋃_{𝔛(F)} 𝔓_{P} ⫋ I(F).

(5) W (F) is connected, i.e., 0 and 1 are the only idempotents in W (F).

(6) W (F) is noetherian if and only if F^{×}/F^{×2} is finite.

Proof. (1): If P ∈ 𝔛(F), then 𝔓_{P} = ⋂_{p} 𝔓_{P,p}, so nil(W (F)) = rad(W (F)). By the Local-Global Principle 31.22, we have

W_{t}(F) = Ker( ∏_{P ∈𝔛(F)} r_{F_{P} /F}) = ⋂_{𝔛(F)} Ker(sgn_{P})

= ⋂_{𝔛(F)} 𝔓_{P} = ⋂_{𝔛(F)} 𝔓_{P,p} = nil(W (F)).

(2): We have sgn_{P} (W (F)^{×}) ⊂ {±1} for all P ∈ 𝔛(F). Let 𝔟 be a nondegenerate symmetric bilinear form satisfying sgn_{P} 𝔟 = ±1 for all P ∈ 𝔛(F). Choose a ∈ F such that 𝔠 := 𝔟 − ⟨a⟩ lies in I^{2}(F) using Proposition 4.13. In particular, sgn_{P} 𝔟 ≡ sgn_{P} ⟨a⟩ mod 4, hence sgn_{P} 𝔟 = sgn_{P} ⟨a⟩ for all P ∈ 𝔛(F). Consequently, sgn_{P} 𝔠 = 0 for all P ∈ 𝔛(F) so 𝔠 is torsion by the Local-Global Principle 31.22. By (1), the form 𝔠 is nilpotent hence 𝔟 ∈ W (F)^{×}.

(3), (4): As the set of zero divisors is a saturated multiplicative set, it follows by commutative algebra that it is a union of prime ideals.

Suppose that F is not pythagorean. Then W_{t}(F) ≠ 0 by Corollary 31.9. In particular, 2^{n}𝔟 = 0 ∈ W (F) for some 𝔟 ≠ 0 in W (F) and n ≥ 1 by Theorem 31.18. Thus ⟨⟨−1⟩⟩ is a zero divisor. As I(F) is the only prime ideal containing ⟨⟨−1⟩⟩, we have I(F) ⊂ zd(W (F)). Since n⟨1⟩ is not a zero divisor for any odd integer n by Theorem 31.18, no 𝔓_{P,p} can lie in zd(W (F)). It follows that zd(W (F)) = I(F), since 𝔓_{P} ⊂ I(F) for all P ∈ 𝔛(F).

Suppose that F is pythagorean. Then W (F) is torsion-free, so n⟨1⟩ is not a zero-divisor for any nonzero integer n. In particular, no maximal ideal lies in zd(W (F)). Let P ∈ 𝔛(F) and 𝔟 ∈ 𝔓_{P}. Then 𝔟 is diagonalizable so we have 𝔟 ≃ ⟨a_{1}, …, a_{n}, b_{1}, … b_{n}⟩ with a_{i}, −b_{j} ∈ P for all i, j. Let 𝔠 = ⟨⟨a_{1}b_{1}, …, a_{n}b_{n}⟩⟩. Then 𝔠 is nonzero in W (F) as sgn_{P} 𝔠 = 2^{n}. As ⟨⟨−a_{i}b_{i}⟩⟩ · 𝔠 = 0 in W (F) for all i, we have 𝔟 · 𝔠 = 0, hence 𝔟 ∈ zd(W (F)). Consequently, 𝔓_{P} ⊂ zd(W (F)) for all P ∈ 𝔛(F), hence zd(W (F)) is the union of the minimal primes.

(5): If the result is false, then 1 = e_{1} + e_{2} for some nontrivial idempotents e_{1}, e_{2}. As e_{1}e_{2} = 0, we have e_{1}, e_{2} ∈ zd(W (F)) ⊂ I(F) which implies 1 ∈ I(F), a contradiction.

(6): This follows by the same proof for the analogous result in Proposition 31.4. □

Let I_{t}(F) := W_{t}(F) ∩ I(F).

Proposition 31.27. If F is formally real, then W_{t}(F) is generated by ⟨⟨x⟩⟩ with x ∈ D(∞⟨1⟩), i.e., W_{t}(F) = I_{t}(F) is generated by torsion 1-fold Pfister forms.

Proof. Let 𝔟 ∈ W_{t}(F). Then 2^{n}𝔟 = 0 for some integer n > 0. Thus 𝔟 ∈ ann_{W (F)}(2^{n}⟨1⟩). By Corollary 6.23, there exist binary forms 𝔡_{i} ∈ ann_{W (F)}(2^{n}⟨1⟩) satisfying 𝔟 = 𝔡_{1} + ⋯ + 𝔡_{m} in W (F). The result follows. □

31.E. The Witt ring up to isomorphism. Because I(F) is the unique ideal of index two in W (F), we can deduce the following due to Cordes and Harrison (cf. [26]):

Theorem 31.28. Let F and K be two fields. Then W (F) and W (K) are isomorphic as rings if and only if W (F)/I^{3}(F) and W (K)/I^{3}(K) are isomorphic as rings.

Proof. The fundamental ideal is the unique ideal of index two in its Witt ring by Theorem 31.24. Therefore, any ring isomorphism W (F) → W (K) induces a ring isomorphism W (F)/I^{3}(F) → W (K)/I^{3}(K).

Conversely, let g : W (F)/I^{3}(F) → W (K)/I^{3}(K) be a ring isomorphism. As the fundamental ideal is the unique ideal of index two in its Witt ring by Theorem 31.24, the map g induces an isomorphism I(F)/I^{3}(F) → I(K)/I^{3}(K). From this and Proposition 4.13, it follows easily that g induces an isomorphism h : F^{×}/F^{×2} → K^{×}/K^{×2}.

We adopt the following notation. For a coset α = xK^{×2}, write ⟨α⟩ and ⟨⟨α⟩⟩ for the forms ⟨x⟩ and ⟨⟨x⟩⟩ in W (K), respectively. We also write s(a) for h(aF^{×2}). Note that s(ab) = s(a)s(b) for all a, b ∈ F^{×}.

By construction,

g(⟨⟨a⟩⟩ + I^{3}(F)) ≡ ⟨⟨s(a)⟩⟩ mod I^{2}(K)/I^{3}(K).

As g(1) = 1, plugging in a = −1, we get ⟨s(−1)⟩ = ⟨−1⟩. In particular,

(31.29)  ⟨s(1)⟩ + ⟨s(−1)⟩ = ⟨1⟩ + ⟨−1⟩ = 0 ∈ W (K).

Since g is a ring homomorphism, we have

g(⟨⟨a, b⟩⟩ + I^{3}(F)) = g(⟨⟨a⟩⟩ + I^{3}(F)) · g(⟨⟨b⟩⟩ + I^{3}(F))
= ⟨⟨s(a)⟩⟩ · ⟨⟨s(b)⟩⟩ + I^{3}(K)
= ⟨⟨s(a), s(b)⟩⟩ + I^{3}(K).

for every a, b ∈ F^{×}.

If a + b ≠ 0, we have ⟨⟨a, b⟩⟩ ≃ ⟨⟨a + b, ab(a + b)⟩⟩ by Lemma 4.15(3). Therefore,

⟨⟨s(a), s(b)⟩⟩ ≡ ⟨⟨s(a + b), s(ab(a + b))⟩⟩ mod I^{3}(K).

By Theorem 6.20, these two 2-fold Pfister forms are equal in W (K). Therefore,

(31.30)  ⟨s(a)⟩ + ⟨s(b)⟩ = ⟨s(a + b)⟩ + ⟨s(ab(a + b))⟩

in W (K).

Let F be the free abelian group with basis the set of isomorphism classes of 1-dimensional forms ⟨a⟩ over F. It follows from Theorem 4.8 and equations (31.29) and (31.30) that the map F → W (K) taking ⟨a⟩ to ⟨s(a)⟩ gives rise to a homomorphism s : W (F) → W (K). Interchanging the roles of F and K, we have, in a similar fashion, a homomorphism W (K) → W (F) which is the inverse of s. □

32. Addendum on torsion

We know by Corollary 6.26 that if 𝔟 ∈ ann_{W (F)}(2⟨1⟩), i.e., if 2𝔟 = 0 in W (F) that 𝔟 ≃ 𝔡_{1} ⊥ ⋯ ⊥ 𝔡_{n} where each 𝔟_{i} is a binary form annihilated by 2. In particular, if 𝔟 is an anisotropic bilinear Pfister form such that 2𝔟 = 0 in W (F), then the pure subform 𝔟′ of 𝔟 satisfies D(𝔟′) ∩ D(2⟨−1⟩) ≠ ∅. In general, if 2^{n}𝔟 = 0 in W (F) with n > 1, then 𝔟 is not isometric to binary forms annihilated by 2^{n} nor does the pure subform of a torsion bilinear Pfister form represent a totally negative element. In this addendum, we construct a counterexample due to Arason and Pfister (cf. [11]). We use the following variant of the Cassels-Pfister Theorem 17.3.

Lemma 32.1. Let char F ≠ 2. Let φ = ⟨a_{1}, …, a_{n}⟩_{q} be anisotropic over F (t) with a_{1}, …, a_{n} ∈ F [t] all satisfying deg a_{i} ≤ 1. Suppose that 0 ≠ q ∈ D(φ_{F (t)}) ∩ F [t]. Then there exist polynomials f_{1}, …, f_{n} ∈ F [t] such that q = φ(f_{1}, …, f_{n}), i.e., F [t] ⊗_{F} φ represents q.

Proof. Let ψ ≃ ⟨−q⟩ ⊥ φ and let
Q := {f = (f_{0}, …, f_{n}) ∈ F [t]^{n+1} | 𝔟_{ψ}(f, f) = 0}.

Choose f ∈ Q such that deg f_{0} is minimal. Assume that the result is false. Then deg f_{0} > 0. Write f_{i} = f_{0}g_{i} + r_{i} with r_{i} = 0 or deg r_{i} < deg f_{0} for each i using the Euclidean Algorithm. So deg r_{i}^{2} ≤ 2 deg f_{0} − 2 for all i. Let g = (1, g_{1}, …, g_{n}) and define h ∈ F [t]^{n+1} by h = cf + dg with c = 𝔟_{ψ}(g, g) and d = −2𝔟_{ψ}(f, g). We have

𝔟_{ψ}(cf + dg, cf + dg) = c^{2}𝔟_{ψ}(f, f) + 2cd𝔟_{ψ}(f, g) + d^{2}𝔟_{ψ}(g, g) = 0,

so h ∈ Q. Therefore,

h_{0} = 𝔟_{ψ}(g, g)f_{0} − 2𝔟_{ψ}(f, g)g_{0} = 𝔟_{ψ}(f_{0}g − 2f, g) = −𝔟_{ψ}(f + r, g),

so

f_{0}h_{0} = −f_{0}𝔟_{ψ}(f + r, g) = −𝔟_{ψ}(f + r, f − r) = 𝔟_{ψ}(r, r) = ∑_{i=1}^{n} a_{i}r_{i},

which is not zero as φ is anisotropic. Consequently,

deg h_{0} + deg f_{0} ≤ max_{i}{deg a_{i}} + 2 deg f_{0} − 2 ≤ 2 deg f_{0} − 1

as deg a_{i} ≤ 1 for all i. This is a contradiction. □

Lemma 32.2. Let F be a formally real field and x, y ∈ D(∞⟨1⟩). Let 𝔟 = ⟨⟨−t, x + ty⟩⟩ be a 2-fold Pfister form over F (t). If 𝔟 ≃ 𝔟_{1} ⊥ 𝔟_{2} over F (t) with 𝔟_{1} and 𝔟_{2} binary torsion forms over F (t), then there exists a z ∈ D(∞⟨1⟩) such that x, y ∈ D(⟨⟨−z⟩⟩).

Proof. If x (respectively, y or xy) is a square, let z = y (respectively, z = x) to finish. So we may assume they are not squares. As 𝔟 is round, we may also assume that 𝔡_{1} ≃ ⟨⟨w⟩⟩ with w ∈ D(∞⟨1⟩) by Corollary 6.6. In particular, D(𝔟′_{F}) ∩ D(∞⟨−1⟩) ≠ ∅ by Lemma 6.11. Thus, there exists a positive integer n such that 𝔟′ ⊥ n⟨1⟩ is isotropic. Let 𝔠 = ⟨t, −(x + yt)⟩ ⊥ n⟨1⟩. We have t(x+yt) ∈ D(𝔠). The form ⟨1, −y⟩ is anisotropic as is n⟨1⟩, since F is formally real. If 𝔠 is isotropic, then we would have an equation −tf^{2} = ∑ g_{i}^{2} − (x + yt)h^{2} in F [t] for some f, g_{i}, h ∈ F [t]. Comparing leading terms implies that y is a square. So 𝔠 is anisotropic. By Lemma 32.1, there exist c, d, f_{i} ∈ F [t] satisfying

f_{1}^{2} + ⋯ + f_{n}^{2} + tc^{2} − (x + yt)d^{2} = t(x + yt).

Since ⟨1, −y⟩ and n⟨1⟩ are anisotropic and t^{2} occurs on the right hand side, we must have c, d are constants and deg f_{i} ≤ 1 for all i. Write f_{i} = a_{i} + b_{i}t with a_{i}, b_{i} ∈ F for 1 ≤ i ≤ n. Then

∑_{i=1}^{n} a_{i}^{2} = xd^{2}, 2 ∑_{i=1}^{n} a_{i}b_{i} = −c^{2} + x + yd^{2}, and ∑_{i=1}^{n} b_{i}^{2} = y.

If d = 0, then a_{i} = 0 for all i and x = c^{2} is a square which was excluded. So d ≠ 0. Let

z = 4 ∑_{i=1}^{n} a_{i}^{2} · ∑_{i=1}^{n} b_{i}^{2} − 4( ∑_{i=1}^{n} a_{i}b_{i})^{2} = 4xyd^{2} − (x − c^{2} + yd^{2})^{2}.

Applying the Cauchy-Schwarz Inequality in each real closure of F, we see that z is nonnegative in every ordering, so z ∈ D̃(∞⟨1⟩). As xy is not a square, z ≠ 0. As d ≠ 0, we have xy ∈ D(⟨⟨−z⟩⟩). Expanding, one checks

z = 4xyd^{2} − (x − c^{2} + yd^{2})^{2} = 4xc^{2} − (x − yd^{2} + c^{2})^{2}.

Thus x ∈ D(⟨⟨−z⟩⟩). As ⟨⟨−z⟩⟩ is round, y ∈ D(⟨⟨−z⟩⟩) also. □

Lemma 32.3. Let F_{0} be a formally real field and u, y ∈ D(∞⟨1⟩_{F_{0}}). Let x = u+t^{2} in F = F_{0}(t). If there exists a z ∈ D(∞⟨1⟩_{F}) such that x, y ∈ D(⟨⟨−z⟩⟩), then y ∈ D(⟨⟨−u⟩⟩).

Proof. We may assume that y is not a square. By assumption, we may write

z = (u + t^{2})f_{1}^{2} − g_{1}^{2} = yf_{2}^{2} − g_{2}^{2} for some f_{1}, f_{2}, g_{1}, g_{2} ∈ F_{0}(t).

Multiplying this equation by an appropriate square in F_{0}(t), we may assume that z ∈ F [t] and that f_{1}, g_{1}, f_{2}, g_{2} ∈ F_{0}[t] have no common nontrivial factor. As z is totally positive, i.e., lies in D(∞⟨1⟩), its leading term must be totally positive in F_{0}. Consequently,

deg g_{1} ≤ 1 + deg f_{1} and deg g_{2} ≤ deg f_{2}.

It follows that \frac{1}{2} deg z ≤ 1 + deg f_{1}. We have \frac{1}{2} deg z = deg f_{2}, otherwise y ∈ F^{2}, a contradiction. Thus, we have

deg f_{2} ≤ 1 + deg f_{1} and deg(g_{1} ± g_{2}) ≤ 1 + deg f_{1}.

If deg((u + t^{2})f_{1}^{2} − yf_{2}^{2}) < 2 deg f_{1} + 2, then y would be a square in F_{0}, a contradiction. So

deg((u + t^{2})f_{1}^{2} − yf_{2}^{2}) = 2 + 2 deg f_{1}.

As (u + t^{2})f_{1}^{2} − yf_{2}^{2} = g_{1}^{2} − g_{2}^{2}, we have deg(g_{1} ± g_{2}) = 1 + deg f_{1}. It follows that either f_{1} or g_{1} − g_{2} has a prime factor p of odd degree. Let \overline{F} = F_{0}[t]/(p) and

: F_{0}[t] → \overline{F} the canonical map. Suppose that \overline{f}_{1} = 0. Then \overline{z} = −\overline{g}_{1}^{2} in \overline{F}. As z is a sum of squares in F_{0}[t] (possibly zero), we must also have \overline{z} is a sum of squares in \overline{F}. But [\overline{F} : F_{0}] is odd, hence \overline{F_{0}} is still formally real by Theorem 95.3 or Springer’s Theorem 18.5. Consequently, we must have \overline{z} = \overline{g}_{1} = 0. This implies that y\overline{f}_{2}^{2} = \overline{g}_{2}^{2}. As y cannot be a square in the odd degree extension \overline{F} of F_{0} by Springer’s Theorem 18.5, we must have \overline{f}_{2} = 0 = \overline{g}_{2}. But there exist no prime p dividing f_{1}, f_{2}, g_{1}, and g_{2}. Thus p |̸ f_{1} in F_{0}[t]. It follows that \overline{g}_{1} = \overline{g}_{2} which in turn implies that (u + \overline{t}^{2})\overline{f}_{1}^{2} − y\overline{f}_{2}^{2} = 0. As \overline{f}_{1} ≠ 0, we have \overline{f}_{2} ≠ 0, so we conclude that ⟨u, 1, −y⟩\overline{F} is isotropic. As [\overline{F} : F_{0}] is odd, ⟨u, 1, −y⟩ is isotropic over F_{0} by Springer’s Theorem 18.5, i.e., y ∈ D(⟨⟨−u⟩⟩) as needed. □

We now construct the counterexample.

Example 32.4. We apply the above lemmas in the following case. Let F_{0} = ℚ(t_{1}) and u = 1 and y = 3. The element y is a sum of three but not two squares in F_{0} by the Substitution Principle 17.7. Let K = F_{0}(t_{2}) and 𝔟 = ⟨⟨−t_{2}, 1 + t_{1}^{2} + 3t_{2}⟩⟩ over K. Then the Pfister form 4𝔟 is isotropic, hence metabolic so 4𝔟 = 0 in W (K). As 1, 3t_{2}^{2} ∈ D(⟨⟨−3t_{2}^{2}⟩⟩_{K}) and 3 ∉ D(2⟨1⟩_{ℚ(t_{1})}), the lemmas imply that 𝔟 is not isometric to an orthogonal sum of binary torsion forms. In particular, it also follows that the form 𝔟 has the property D(𝔟′) ∩ D(∞⟨−1⟩_{K}) = ∅.

33. The total signature

We saw when F is a formally real field, the torsion in the Witt ring W (F) is determined by the signatures at the orderings on F. In this section, we view the relationship between bilinear forms over a formally real field F and the totality of continuous functions on the topological space 𝔛(F) of orderings on F with integer values.

We shall use results in Appendices §95 and §96. Let F be a formally real field. The space of orderings 𝔛(F) is a boolean space, i.e., a totally disconnected compact Hausdorff space with a subbase the collection of sets

(33.1)  H(a) = H_{F} (a) := {P ∈ 𝔛(F) | −a ∈ P }.

Let 𝔟 be a nondegenerate symmetric bilinear form over F. Then we define the total signature of 𝔟 to be the map

(33.2)  sgn 𝔟 : 𝔛(F) → ℤ given by sgn 𝔟(P) = sgn_{P} 𝔟.

Theorem 33.3. Let F be formally real. Then

sgn 𝔟 : 𝔛(F) → ℤ

is continuous with respect to the discrete topology on ℤ. The topology on 𝔛(F) is the coarsest topology such that sgn 𝔟 is continuous for all 𝔟.

Proof. As ℤ is a topological group, addition of continuous functions is continuous. As any nondegenerate symmetric bilinear form is diagonalizable over a formally real field, we need only prove the result for 𝔟 = ⟨a⟩, a ∈ F^{×}. But

(sgn⟨a⟩)^{−1}(n) = \{ \begin{matrix}∅ & if n ≠ ±1, \\ H(a) & if n = −1, \\ H(−a) & if n = 1. \end{matrix}

The result follows easily as the H(a) form a subbase. □

Let C(𝔛(F), ℤ) be the ring of continuous functions f : 𝔛(F) → ℤ where ℤ has the discrete topology. By the theorem, we have a map

(33.4)  sgn : W (F) → C(𝔛(F), ℤ) given by 𝔟 ↦ sgn 𝔟

called the total signature map. It is a ring homomorphism. The Local-Global Theorem 31.22 in this terminology states

W_{t}(F) = Ker(sgn).

We turn to the cokernel of sgn : W (F) → C(𝔛(F), ℤ). We shall show that it too is a 2-primary torsion group. This generalizes the two observations that C(𝔛(F), ℤ) = 0 if F is not formally real and sgn : W (F) → C(𝔛(F), ℤ) is an isomorphism if F is euclidean. We use and generalize the approach and results from [33] that we shall need in §41.

If A ⊂ 𝔛(F), write χ_{A} for the characteristic function of A. In particular, χ_{A} ∈ C(𝔛(F), ℤ) if A is clopen. Let f ∈ C(𝔛(F), ℤ). Then A_{n} = f^{−1}(n) is a clopen set. As {A_{n} | n ∈ ℤ} partition the compact space 𝔛(F), only finitely many A_{n} are nonempty. In particular, f = ∑ nχ_{A_{n}} is a finite sum. This shows that C(𝔛(F), ℤ) is additively generated by χ_{A}, as A varies over the clopen sets in the boolean space 𝔛(F).

The finite intersections of the subbase elements (33.1),

(33.5)  H(a_{1}, …, a_{n}) := H(a_{1}) ∩ ⋯ ∩ H(a_{n}) with a_{1}, …, a_{n} ∈ F^{×}

form a base for the topology of 𝔛(F). As

H(a_{1}, …, a_{n}) = supp(⟨⟨a_{1}, …, a_{n}⟩⟩),

where supp 𝔟 := {P ∈ 𝔛(F) | sgn_{P} 𝔟 ≠ 0} is the support of 𝔟, this base is none other than the collection of clopen sets

(33.6)  {supp(𝔟) | 𝔟 is a bilinear Pfister form}.

We also have

(33.7)  sgn 𝔟 = 2^{n}χ_{supp(𝔟)} if 𝔟 is a bilinear n-fold Pfister form.

Theorem 33.8. The cokernel of sgn : W (F) → C(𝔛(F), ℤ) is 2-primary torsion.

Proof. It suffices to prove for each clopen set A ⊂ 𝔛(F) that 2^{n}χ_{A} ∈ im(sgn) for some n ≥ 0. As 𝔛(F) is compact, A is a finite union of clopen sets of the form (33.6) whose characteristic functions lie in im(sgn) by (33.7). By induction, it suffices to show that if A and B are clopen sets in 𝔛(F) with 2^{n}χ_{A} and 2^{m}χ_{B} lying in im(sgn) for some integers m and n, then 2^{s}χ_{A∪B} lies in im(sgn) for some s. But

(33.9)  χ_{A∪B} = χ_{A} + χ_{B} − χ_{A} · χ_{B},

so

(33.10)  2^{m+n}χ_{A∪B} = 2^{m}(2^{n}χ_{A}) + 2^{n}(2^{m}χ_{B}) − (2^{n}χ_{A}) · (2^{m}χ_{B})

lies in im(sgn) as needed. □

Define the reduced stability index st_{r}(F) of F to be n if 2^{n} is the exponent of the cokernel of the signature map (or infinity if this exponent is not finite).

Refining the argument in the last theorem, we establish:

Lemma 33.11. Let C ⊂ 𝔛(F) be clopen. Then there exists an integer n > 0 and a 𝔟 ∈ I^{n}(F) satisfying sgn 𝔟 = 2^{n}χ_{C}. More precisely, there exists an integer n > 0, bilinear n-fold Pfister forms 𝔟_{i} satisfying supp(𝔟_{i}) ⊂ C, and integers k_{i} such that ∑ k_{i} sgn 𝔟_{i} = 2^{n}χ_{C}.

Proof. As 𝔛(F) is compact and (33.6) is a base for the topology, there exists an r ≥ 1 such that C = A_{1} ∪ ⋯ ∪ A_{r} with A_{i} = supp(𝔟_{i}) for some m_{i}-fold Pfister forms 𝔟_{i}, i ∈ [1, r]. We induct on r. If r = 1 the result follows by (33.7), so assume that r > 1. Let A = A_{1}, 𝔟 = 𝔟_{1}, and B = A_{2} ∪ ⋯ ∪ A_{r}. By induction, there exists an m ≥ 1 and a 𝔠 ∈ I^{m}(F), a sum (and difference) of Pfister forms with the desired properties with sgn 𝔠 = 2^{m}χ_{B}. Multiplying by a suitable power of 2, we may assume that m = m_{1}. Let 𝔡 = 2^{m}(𝔟 ⊥ 𝔠) ⊥ (−𝔟) ⊗ 𝔠. Then 𝔡 is a sum (and difference) of Pfister forms whose supports all lie in C as supp(𝔞) = supp(2𝔞) for any bilinear form 𝔞. By equations (33.9) and (33.10), we have

2^{2m}χ_{A∪B} = 2^{2m}χ_{A} + 2^{2m}χ_{B} − 2^{m}χ_{A} · 2^{m}χ_{B}
= 2^{m}(sgn 𝔟 + sgn 𝔠) − sgn 𝔟 · sgn 𝔠 = sgn 𝔡,

so the result follows. □

Using the lemma, we can establish two useful results. The first is:

Theorem 33.12 (Normality Theorem). Let A and B be disjoint closed subsets of 𝔛(F). Then there exists an integer n > 0 and 𝔟 ∈ I^{n}(F) satisfying

sgn_{P} 𝔟 = \begin{cases}2^{n} & if P ∈ A, \\ 0 & if P ∈ B. \end{cases}

Proof. The complement 𝔛(F) \ B is a union of clopen sets. As the closed set A is covered by this union of clopen sets and 𝔛(F) is compact, there exists a finite cover {C_{1}, …, C_{r}} of A for some clopen sets C_{i}, i ∈ [1, r] lying in 𝔛(F) \ B. As C_{i} \ (C_{i} ∩ (⋃_{j≠i} C_{j})) is clopen for i ∈ [1, r], we may assume this is a disjoint union. By Lemma 33.11, there exist 𝔟_{i} ∈ I^{m_{i}}(F), some m_{i}, such that sgn 𝔟_{i} = 2^{m_{i}}χ_{C_{i}}. Let n = max{m_{i} | 1 ≤ i ≤ r}. Then 𝔟 = ∑_{i} 2^{n−m_{i}}𝔟_{i} lies in I^{n}(F) and satisfies 𝔟 = 2^{n}χ_{∪_{i}C_{i}}. Since A ⊂ ∪_{i}C_{i}, the result follows. □

We now investigate the relationship between elements in f ∈ C(𝔛(F), 2^{m}ℤ) and bilinear forms 𝔟 satisfying 2^{m} | sgn_{P} 𝔟 for all P ∈ 𝔛(F). We first need a useful trick.

If ε = (ε_{1}, …, ε_{n}) ∈ {±1}^{n} and 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ with a_{i} ∈ F^{×}, let
𝔟_{ε} = ⟨⟨ε_{1}a_{1}, …, ε_{n}a_{n}⟩⟩.

Then supp(𝔟_{ε}) ∩ supp(𝔟_{ε′)} = ∅ unless ε = ε′.

Lemma 33.13. Let 𝔟 be a bilinear n-fold Pfister form over an arbitrary field F. Then 2^{n}⟨1⟩ = ∑_{ε} 𝔟_{ε} in W (F), where the sum runs over all ε ∈ {±1}^{n}.

Proof. Let 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩ and 𝔠 = ⟨⟨a_{1}, …, a_{n−1}⟩⟩ with a_{i} ∈ F^{×}. As ⟨⟨−1⟩⟩ = ⟨⟨a⟩⟩ + ⟨⟨−a⟩⟩ in W (F) for all a ∈ F^{×}, we have

∑_{ε} 𝔟_{ε} = ∑_{ε′} 𝔠_{ε′}⟨⟨a_{n}⟩⟩ + ∑_{ε′} 𝔠_{ε′}⟨⟨−a_{n}⟩⟩ = 2 ∑_{ε′} 𝔠_{ε′}

where the ε′ run over all {±1}^{n−1}. The result follows by induction on n. □

The generalization of Lemma 33.13 given in the following exercise is useful. These identities were first observed by Witt, who used them to give a simple proof that W_{t}(F) is 2-primary (cf. Exercise 31.21(3)).

Exercise 33.14. Let a_{1}, …, a_{n} ∈ F^{×} and ε = (ε_{1}, …, ε_{n}) ∈ {±1}^{n}. Suppose that 𝔟 = ⟨⟨a_{1}, …, a_{n}⟩⟩, 𝔠 = ⟨−a_{1}, …, −a_{n}⟩, and 𝔢_{ε} = ⟨ε_{1}, …, ε_{n}⟩. Then the following are true in W (F):

(1) 𝔠 · 𝔟_{ε} = 𝔢_{ε} · 𝔟_{ε} = (∑_{i} ε_{i})𝔟_{ε}.

(2) 2^{n}𝔠 = ∑_{ε} 𝔢_{ε} · 𝔟_{ε}.

Using Lemma 33.11, we also establish:

Theorem 33.15. Let f ∈ C(𝔛(F), 2^{m}ℤ). Then there is a positive integer n and a 𝔟 ∈ I^{m+n}(F) such that 2^{n}f = sgn 𝔟. More precisely, there exists an integer n such that 2^{n}f can be written as a sum ∑_{i=1}^{r} k_{i} sgn 𝔟_{i} for some integers k_{i} and bilinear (n + m)-fold Pfister forms 𝔟_{i} such that supp(𝔟_{i}) ⊂ supp(f) for every i = 1, …, r and whose supports are pairwise disjoint.

Proof. We first show:

Claim: Let g ∈ C(𝔛(F), ℤ). Then there exists a nonnegative integer n and bilinear n-fold Pfister forms 𝔠_{i} such that 2^{n}g = ∑_{i=1}^{r} s_{i} sgn 𝔠_{i} for some integers s_{i} with supp(𝔠_{i}) ⊂ supp(g) for every i = 1, …, r.

The function g is a finite sum of functions ∑_{i} iχ_{g^{−1}(i)} with i ∈ ℤ and each g^{−1}(i) a clopen set. For each nonempty g^{−1}(i), there exist a nonnegative integer n_{i}, bilinear n_{i}-fold Pfister forms 𝔟_{ij} with supp(𝔟_{ij}) ⊂ g^{−1}(i) and integers k_{j} satisfying 2^{n_{i}}χ_{g^{−1}(i)} = ∑_{j} k_{j} sgn 𝔟_{ij} by Lemma 33.11. Let n = max_{i}{n_{i}}. Then 2^{n}g = ∑_{i,j} ik_{j} sgn(2^{n−n_{i}}𝔟_{ij}). This proves the Claim.

Let g = f/2^{m}. By the Claim, 2^{n}g = ∑_{i=1}^{r} s_{i} sgn 𝔠_{i} for some n-fold Pfister forms 𝔠_{i} whose supports lie in supp(g) = supp(f). Thus 2^{n}f = ∑_{i=1}^{r} s_{i} sgn 2^{m}𝔠_{i} with each 2^{m}𝔠_{i} an (n+m)-fold Pfister form. Let 𝔡 = 𝔠_{1}⊗⋯⊗𝔠_{r} be an rn-fold Pfister form. By Lemma 33.13, we have 2^{(n+1)r}f = ∑_{ε} sgn(2^{m}s_{i}𝔠_{i} · 𝔡_{ε}) in C(𝔛(F), ℤ) where ε runs over all {±1}^{rn}. For each i and ε, the form 𝔠_{i} · 𝔡_{ε} is isometric to either 2^{n+m}𝔡_{ε} or is metabolic by Example 4.16(2) and (3). As the 𝔡_{ε} have pairwise disjoint supports, adding the coefficients of the isometric forms 𝔠_{i} · 𝔡_{ε} yields the result. □

Corollary 33.16. Let 𝔟 be a nondegenerate symmetric bilinear form over F and fix m > 0. Then 2^{n}𝔟 ∈ I^{n+m}(F) for some n ≥ 0 if and only if

sgn 𝔟 ∈ C(𝔛(F), 2^{m}ℤ).

Proof. We may assume that F is formally real as 2s(F)W (F) = 0.

⇒: If 𝔡 is a bilinear n-fold Pfister form, then sgn 𝔡 ∈ C(𝔛(F), 2^{n}ℤ). It follows that sgn(I^{n}(F)) ⊂ C(𝔛(F), 2^{n}ℤ). Suppose that 2^{n}𝔟 ∈ I^{n+m}(F) for some n ≥ 0. Then 2^{n} sgn 𝔟 ∈ C(𝔛(F), 2^{n+m}ℤ), hence sgn 𝔟 ∈ C(𝔛(F), 2^{m}ℤ).

⇐: By Theorem 33.15, there exists 𝔠 ∈ I^{n+m}(F) such that sgn 𝔠 = 2^{n} sgn 𝔟. As W_{t}(F) = Ker(sgn) is 2-primary torsion by the Local-Global Principle 31.22, there exists a nonnegative integer k such that 2^{n+k}𝔟 = 2^{k}𝔠 ∈ I^{n+m+k}(F). □

This Corollary 33.16 suggests that if 𝔟 is a nondegenerate symmetric bilinear form over F, the following may be true:

(33.17)  sgn 𝔟 ∈ C(𝔛(F), 2^{n}ℤ) if and only if 𝔟 ∈ I^{n}(F) + W_{t}(F).

This question was raised by Lam in [88].

In particular, in the case that F is a formally real pythagorean field, this would mean

𝔟 ∈ I^{n}(F) if and only if 2^{n} | sgn_{P} (𝔟) for all P ∈ 𝔛(F)

as W (F) is then torsion-free which would answer a question of Marshall in [97].

Of course, if 𝔟 ∈ I^{n}(F) + W_{t}(F), then sgn 𝔟 ∈ C(𝔛(F), 2^{n}ℤ). The converse would follow if

2^{m}𝔟 ∈ I^{n+m}(F) always implies that 𝔟 ∈ I^{n}(F) + W_{t}(F).

If F were formally real pythagorean, the converse would follow if

2^{m}𝔟 ∈ I^{n+m}(F) always implies that 𝔟 ∈ I^{n}(F).

Because the nilradical of W (F) is the torsion W_{t}(F) when F is formally real, the total signature induces an embedding of the reduced Witt ring

W_{red}(F) := W (F)/nil(W (F)) = W (F)/W_{t}(F)

into C(𝔛(F), ℤ). Moreover, since W_{t}(F) is 2-primary, the images of two nondegenerate bilinear forms 𝔟 and 𝔠 are equal in the reduced Witt ring if and only if there exists a nonnegative integer n such that 2^{n}𝔟 = 2^{n}𝔠 in W (F). Let : W (F) → W_{red}(F) be the canonical ring epimorphism. Then the problem above becomes: If 𝔟 is a nondegenerate symmetric bilinear form over F, then

\overline{𝔟} ∈ I_{red}^{n}(F) if and only if sgn 𝔟 ∈ C(𝔛(F), 2^{n}ℤ).

where I_{red}^{n}(F) is the image of I^{n}(F) in W_{red}(F).

This is all, in fact, true as we shall see in §41 (Cf. Corollaries 41.9 and 41.10).

34. Bilinear and quadratic forms under quadratic extensions

In this section we develop the relationship between bilinear and quadratic forms over a field F and over a quadratic extension K of F. We know that bilinear and quadratic forms can become isotropic over a quadratic extension and we exploit this. We also investigate the transfer map taking forms over K to forms over F induced by a nontrivial F -linear functional. This leads to useful exact sequences of Witt rings and Witt groups done first in the case of characteristic not 2 in [37] and by Arason in [4] in the case of characteristic 2 by Baeza in [15].

34.A. Bilinear forms under a quadratic extension. We start with bilinear forms and the Witt ring.

Proposition 34.1. Let K/F be a quadratic field extension and s : K → F a nontrivial F -linear functional satisfying s(1) = 0. Let 𝔠 be an anisotropic bilinear form over K. Then there exist bilinear forms 𝔟 over F and 𝔞 over K such that 𝔠 ≃ 𝔟_{K} ⊥ 𝔞 with s_{∗}(𝔞) anisotropic.

Proof. We induct on dim 𝔠. Suppose that s_{∗}(𝔠) is isotropic. It follows that there is a b ∈ D(𝔠) ∩ F, i.e., 𝔠 ≃ ⟨b⟩ ⊥ 𝔠_{1} for some 𝔠_{1}. Applying the induction hypothesis to 𝔠_{1} completes the proof. □

We need the following generalization of Proposition 34.1.

Lemma 34.2. Let K/F be a quadratic extension of F and s : K → F a nontrivial F -linear functional satisfying s(1) = 0. Let 𝔣 be a bilinear anisotropic n-fold Pfister form over F and 𝔠 a nondegenerate bilinear form over K such that 𝔣_{K} ⊗ 𝔠 is anisotropic. Then there exists a bilinear form 𝔟 over F and a bilinear form 𝔞 over K such that 𝔣_{K} ⊗ 𝔠 ≃ (𝔣 ⊗ 𝔟)_{K} ⊥ 𝔣_{K} ⊗ 𝔞 with 𝔣 ⊗ s_{∗}(𝔞) anisotropic.

Proof. Let 𝔡 = 𝔣_{K} ⊗ 𝔠. We may assume that s_{∗}(𝔡) is isotropic. Then there exists a b ∈ D(𝔡) ∩ F. If 𝔠 ≃ ⟨a_{1}, …, a_{n}⟩, there exist x_{i} ∈ D̃(𝔣_{K}), not all zero satisfying b = x_{1}a_{1} + ⋯ + x_{n}a_{n}. Let y_{i} = x_{i} if x_{i} ≠ 0 and y_{i} = 1 otherwise. Then

𝔣_{K} ⊗ 𝔠 ≃ 𝔣_{K} ⊗ ⟨y_{1}a_{1}, …, y_{n}a_{n}⟩ ≃ 𝔣_{K} ⊗ ⟨b, z_{2}, …, z_{n}⟩

for some z_{i} ∈ K^{×} as G(𝔣_{K}) = D(𝔣_{K}). The result follows easily by induction. □

Corollary 34.3. Let K/F be a quadratic extension of F and s : K → F a nontrivial F -linear functional satisfying s(1) = 0. Let 𝔣 be a bilinear anisotropic n-fold Pfister form and 𝔠 an anisotropic bilinear form over K satisfying 𝔣 ⊗ s_{∗}(𝔠) is hyperbolic. Then there exists a bilinear form 𝔟 over F such that dim 𝔟 = dim 𝔠 and 𝔣_{K} ⊗ 𝔠 ≃ (𝔣 ⊗ 𝔟)_{K}.

Proof. If 𝔣_{K} ⊗ 𝔠 is anisotropic, the result follows by Lemma 34.2, so we may assume that 𝔣_{K} ⊗ 𝔠 is isotropic. If 𝔣_{K} is isotropic, it is hyperbolic and the result follows easily, so we may assume the Pfister form 𝔣_{K} is anisotropic. Using Proposition 6.22, we see that there exist a bilinear form 𝔡 with 𝔣_{K} ⊗ 𝔡 anisotropic and an integer n ≥ 0 with dim 𝔡 + 2n = dim 𝔠 and 𝔣_{K} ⊗ 𝔠 ≃ 𝔣_{K} ⊗ (𝔡 ⊥ nℍ). Replacing 𝔠 by 𝔡, we reduce to the anisotropic case. □

Note that if K/F is a quadratic extension and s, s′ : K → F are F -linear functionals satisfying s(1) = 0 = s′(1) with s nontrivial, then s′_{∗} = as_{∗} for some a ∈ F.

Theorem 34.4. Let K/F be a quadratic field extension and s : K → F a nonzero F -linear functional such that s(1) = 0. Then the sequence

W (F) ⟶^{r_{K/F}} W (K) ⟶^{s_{∗}} W (F)

is exact.

Proof. Let b ∈ F^{×} then the binary form s_{∗}(⟨b⟩_{K}) is isotropic, hence metabolic. Thus s_{∗} ∘ r_{K/F} = 0. Let 𝔠 ∈ W (K). By Proposition 34.1, there exists a decomposition 𝔠 ≃ 𝔟_{K} ⊥ 𝔠_{1} with 𝔟 a bilinear form over F and 𝔠_{1} a bilinear form over K satisfying s_{∗}(𝔠_{1}) is anisotropic. In particular, if s_{∗}(𝔠) = 0, we have 𝔠 = 𝔟_{K}. This proves exactness. □

If K/F is a quadratic extension, denote the quadratic norm form of the quadratic algebra K by N_{K/F} (cf. §98.B).

Lemma 34.5. Let K/F be a quadratic extension and s : K → F a nontrivial F - linear functional. Let 𝔟 be an anisotropic binary bilinear form over F such that the quadratic form 𝔟 ⊗ N_{K/F} is isotropic. Then 𝔟 ≃ s_{∗}(⟨y⟩) for some y ∈ K^{×}.

Proof. Let {1, x} be a basis of K over F. Let 𝔠 be the polar form of N_{K/F}. We have

𝔠(1, x) = N_{K/F} (1 + x) − N_{K/F} (x) − N_{K/F} (1) = Tr_{K/F} (x)

for every x ∈ K. By assumption there are nonzero vectors v, w ∈ V_{𝔟} such that

0 = (𝔟 ⊗ N_{K/F})(v ⊗ 1 + w ⊗ x)
= 𝔟(v, v) N_{K/F} (1) + 𝔟(v, w)𝔠(1, x) + 𝔟(w, w) N_{K/F} (x)
= 𝔟(v, v) + 𝔟(v, w) Tr_{K/F} (x) + 𝔟(w, w) N_{K/F} (x)

by the definition of tensor product (8.14). Let f : K → F be an F -linear functional satisfying f(1) = 𝔟(w, w) and f(x) = 𝔟(v, w). By (98.1), we have

f(x^{2}) = f(− Tr_{K/F} (x)x − N_{K/F} (x))

= − Tr_{K/F} (x)𝔟(v, w) − N_{K/F} (x)𝔟(w, w) = 𝔟(v, v).

Therefore, the F -linear isomorphism K → V_{𝔟} taking 1 to w and x to v is an isometry between 𝔠 = f_{∗}(⟨1⟩) and 𝔟. As f is the composition of s with the endomorphism of K given by multiplication by some element y ∈ K^{×}, we have 𝔟 ≃ f_{∗}(⟨1⟩) ≃ s_{∗}(⟨y⟩). □

Proposition 34.6. Let K/F be a quadratic extension and s : K → F a nontrivial F -linear functional. Let 𝔟 be an anisotropic bilinear form over F. Then there exist bilinear forms 𝔠 over K and 𝔡 over F such that 𝔟 ≃ s_{∗}(𝔠) ⊥ 𝔡 and 𝔡 ⊗ N_{K/F} is anisotropic.

Proof. We induct on dim 𝔟. Suppose that 𝔟 ⊗ N_{K/F} is isotropic. Then there is a 2-dimensional subspace W ⊂ V_{𝔟} with (𝔟|_{W})⊗N_{K/F} isotropic. By Lemma 34.5, we have 𝔟|_{W} ≃ s_{∗}(⟨y⟩) for some y ∈ K^{×}. Applying the induction hypothesis to the orthogonal complement of W in V completes the proof. □

Theorem 34.7. Let K = F (\sqrt{a}) be a quadratic field extension of F with a ∈ F^{×}. Let s : K → F be a nontrivial F -linear functional such that s(1) = 0. Then the sequence

W (K) ⟶^{s_{∗}} W (F) ⟶^{⟨⟨a⟩⟩} W (F)

is exact where the last homomorphism is multiplication by ⟨⟨a⟩⟩.

Proof. For every 𝔠 ∈ W (F) we have ⟨⟨a⟩⟩s_{∗}(𝔠) = s_{∗}(⟨⟨a⟩⟩_{K}𝔠) = 0 as ⟨⟨a⟩⟩_{K} = 0. Therefore, the composition of the two homomorphisms in the sequence is trivial. Since N_{K/F} ≃ ⟨⟨a⟩⟩_{q}, the exactness of the sequence now follows from Proposition 34.6. □

34.B. Quadratic forms under a quadratic extension. We now turn to quadratic forms.

Proposition 34.8. Let K/F be a separable quadratic field extension and φ an anisotropic quadratic form over F. Then

φ ≃ (𝔟 ⊗ N_{K/F}) ⊥ ψ

with 𝔟 a nondegenerate symmetric bilinear form and ψ a quadratic form satisfying ψ_{K} is anisotropic.

Proof. Since K/F is separable, the binary form σ := N_{K/F} is nondegenerate. As F (σ) ≃ K, the statement follows from Corollary 22.12. □

Theorem 34.9. Let K/F be a separable quadratic field extension and s : K → F a nonzero functional such that s(1) = 0. Then the sequence

W (F) ⟶^{r_{K/F}} W (K) ⟶^{s_{∗}} W (F) ⟶^{N_{K/F}} I_{q}(F) ⟶^{r_{K/F}} I_{q}(K) ⟶^{s_{∗}} I_{q}(F)

is exact where the middle homomorphism is multiplication by N_{K/F}.

Proof. In view of Theorem 34.4 and Propositions 34.6 and 34.8, it suffices to prove exactness at I_{q}(K). Let φ ∈ I_{q}(K) be an anisotropic form such that s_{∗}(φ) is hyperbolic. We show by induction on n = dim_{K} φ that φ ∈ im(r_{K/F}). We may assume that n > 0. Let W ⊂ V_{φ} be a totally isotropic F -subspace for the form s_{∗}(φ) of dimension n. As Ker(s) = F we have φ(W) ⊂ F.

We claim that the K-space KW properly contains W, in particular,

(34.10)  dim_{K} KW = \frac{1}{2} dim_{F} KW > \frac{1}{2} dim_{F} W = \frac{n}{2}.

To prove the claim choose an element x ∈ K such that x^{2} ∉ F. Then for every nonzero w ∈ W, we have φ(xw) = x^{2}φ(w) ∉ F, hence xw ∈ KW but x ∉ W. It follows from the inequality (34.10) that the restriction of 𝔟_{φ} on KW and therefore on W is nonzero. Consequently, there is a 2-dimensional F -subspace U ⊂ W such that 𝔟_{φ}|_{U} is nondegenerate. Therefore, the K-space KU is also 2-dimensional and the restriction ψ = φ|_{U} is a nondegenerate binary quadratic form over F satisfying ψ_{K} ≃ φ|_{KU}. Applying the induction hypothesis to (ψ_{K})^{⊥}, we have (ψ_{K})^{⊥} ∈ im(r_{K/F}). Therefore, φ = ψ_{K} + (ψ_{K})^{⊥} ∈ im(r_{K/F}). □

Remark 34.11. In Proposition 34.9, we have Ker(r_{K/F} : I_{q}(F) → I_{q}(K)) = W (F)⟨⟨a]] when K = F_{a}.

Application of the above in the case of fields of characteristic not 2 provides a proof of the following result shown in [37] and by Arason in [4]:

Corollary 34.12. Suppose that char F ≠ 2 and K = F (\sqrt{a})/F is a quadratic field extension with a ∈ F^{×}. If s : K → F is a nontrivial F -linear functional such that s(1) = 0, then the triangle

\xymatrix{ & W (K) \ar[dr]^{s_{∗}} \\ W (F) \ar[ur]^{r_{K/F}} &  & W (F) \ar[ll]_{·⟨⟨a⟩⟩}}

is exact.

Proof. Since the quadratic norm form N_{K/F} coincides with φ_{𝔟} where 𝔟 = ⟨⟨a⟩⟩, the map W (F) → I_{q}(F) given by multiplication by N_{K/F} is identified with the map W (F) → I(F) given by multiplication by ⟨⟨a⟩⟩. Note also that Ker(r_{K/F}) ⊂ I(F), so the statement follows from Theorem 34.9. □

Remark 34.13. Suppose that char F ≠ 2 and K = F (\sqrt{a}) is a quadratic extension of F. Let 𝔟 be an anisotropic bilinear form. Then by Proposition 34.8 and Example 9.4, we see that the following are equivalent:

(1) 𝔟_{K} is metabolic.

(2) 𝔟 ∈ ⟨⟨a⟩⟩W (F).

(3) 𝔟 ≃ ⟨⟨a⟩⟩ ⊗ 𝔠 for some symmetric bilinear form 𝔠.

In the case that char F = 2, Theorem 34.9 can be slightly improved.

We need the following computation:

Lemma 34.14. Let F be a field of characteristic 2 and K/F a quadratic field extension. Let s : K → F be a nonzero F -linear functional satisfying s(1) = 0. Then for every x ∈ K we have

s_{∗}(⟨⟨x]]) = \{ \begin{matrix}0 & if x ∈ F, \\ s(x)⟨⟨Tr_{K/F} (x)]] & otherwise. \end{matrix}

In particular, s_{∗}(⟨⟨x]]) ≡ ⟨⟨Tr_{K/F} (x)]] modulo I_{q}^{2}(F).

Proof. The element x satisfies the quadratic equation x^{2} + ax + b = 0 for some a, b ∈ F. We have Tr_{K/F} (x) = a and s(x^{2}) = as(x) = s(x) Tr_{K/F} (x). Let x̄ = Tr_{K/F} (x) − x. The element x̄ satisfies the same quadratic equation and s(x̄^{2}) = s(x) Tr_{K/F} (x̄).

Let {v, w} be the standard basis for the space V of the form φ := ⟨⟨x]] over K. If x ∈ F, then v and w span the totally isotropic F -subspace of s_{∗}(φ), i.e., s_{∗}(φ) = 0.

Suppose that x ∉ F. Then V = W ⊥ W′ with W = F v ⊕ F xw and W′ = F x̄v ⊕ F w. We have s_{∗}(φ) ≃ s_{∗}(φ)|_{W} ⊥ s_{∗}(φ)|_{W′}. As s_{∗}(φ)(v) = s(1) = 0, the form s_{∗}(φ)|_{W} is isotropic and therefore s_{∗}(φ)|_{W} ≃ ℍ. Moreover,

s_{∗}(φ)(x̄v) = s(x̄^{2}) = s(x) Tr_{K/F} (x), s_{∗}(φ)(w) = s(x) and
s_{∗}(𝔟_{φ}(x̄v, w)) = s(x̄) = s(x),

hence s_{∗}(φ)|_{W′} ≃ s(x)⟨⟨Tr_{K/F} (x)]]. □

Corollary 34.15. Suppose that char F = 2. Let K/F be a separable quadratic field extension and s : K → F a nonzero functional such that s(1) = 0. Then the sequence

0 → W (F) ⟶^{r_{K/F}} W (K) ⟶^{s_{∗}} W (F) ⟶^{· N_{K/F}} I_{q}(F) ⟶^{r_{K/F}} I_{q}(K) ⟶^{s_{∗}} I_{q}(F) → 0

is exact.

Proof. To prove the injectivity of r_{K/F}, it suffices to show that if 𝔟 is an anisotropic bilinear form over F, then 𝔟_{K} is also anisotropic. Let x ∈ K \ F be an element satisfying x^{2} + x + a = 0 for some a ∈ F and let 𝔟_{K}(v + xw, v + xw) = 0 for some v, w ∈ V_{𝔟}. We have

0 = 𝔟_{K}(v + xw, v + xw) = 𝔟(v, v) + a𝔟(w, w) + x𝔟(w, w),

hence 𝔟(w, w) = 0 = 𝔟(v, v). Therefore, v = w = 0 as 𝔟 is anisotropic.

By Lemma 34.14, we have for every y ∈ K, the form s_{∗}(⟨⟨y]]) is similar to ⟨⟨Tr_{K/F} (y)]]. As the map s_{∗} is W (F)-linear, I_{q}(F) is generated by the classes of binary forms and the trace map Tr_{K/F} is surjective, the last homomorphism s_{∗} in the sequence is surjective. □

34.C. The filtrations of the Witt ring and Witt group under quadratic extensions. We turn to the study of relations between the ideals I^{n}(F), I^{n}(K) and between the groups I_{q}^{n}(F) and I_{q}^{n}(K) for a quadratic field extension K/F.

Lemma 34.16. Let K/F be a quadratic extension. Let n ≥ 1.

(1) We have

I^{n}(K) = I^{n−1}(F)I(K),

i.e., I^{n}(K) is the W (F)-module generated by n-fold bilinear Pfister forms 𝔟_{K} ⊗ ⟨⟨x⟩⟩ with x ∈ K^{×} and 𝔟 an (n − 1)-fold bilinear Pfister form over F.

(2) If char F = 2, then

I_{q}^{n}(K) = I^{n−1}(F)I_{q}(K) + I(K)I_{q}^{n−1}(F).

Proof. (1): Clearly, to show that I^{n}(K) = I^{n−1}(F)I(K), it suffices to show this for the case n = 2. Let x, y ∈ K \ F. As 1, x, y are linearly dependent over F, there are a, b ∈ F^{×} such that ax + by = 1. Note that the form ⟨⟨ax, by⟩⟩ is isotropic and therefore metabolic. Using the relation

⟨⟨uv, w⟩⟩ = ⟨⟨u, w⟩⟩ + u⟨⟨v, w⟩⟩

in W (K), we have

0 = ⟨⟨ax, by⟩⟩ = ⟨⟨x, by⟩⟩ + a⟨⟨x, by⟩⟩ = ⟨⟨a, b⟩⟩ + b⟨⟨a, y⟩⟩ + a⟨⟨x, b⟩⟩ + ab⟨⟨x, y⟩⟩,

hence ⟨⟨x, y⟩⟩ ∈ I(F)I(K).

(2): In view of (1), it is sufficient to consider the case n = 2. The group I_{q}^{2}(K) is generated by the classes of 2-fold Pfister forms by (9.5). Let x, y ∈ K. If x ∈ F, then ⟨⟨x, y]] ∈ I(F)I_{q}(K). Otherwise, y = a + bx for some a, b ∈ F. Then, by Lemma 15.1 and Lemma 15.5,

⟨⟨x, y]] = ⟨⟨x, a]] + ⟨⟨x, bx]] = ⟨⟨x, a]] + ⟨⟨b, bx]] ∈ I(K)I_{q}(F) + I(F)I_{q}(K)

since ⟨⟨b, bx]] + ⟨⟨x, bx]] = ⟨⟨bx, bx]] = 0. □

Corollary 34.17. Let K/F be a quadratic extension and s : K → F a nonzero F -linear functional. Then for every n ≥ 1:

(1) s_{∗}(I^{n}(K)) ⊂ I^{n}(F).

(2) s_{∗}(I_{q}^{n}(K)) ⊂ I_{q}^{n}(F).

Proof. (1): Clearly, s_{∗}(I(K)) ⊂ I(F). It follows from Lemma 34.16 and Frobenius Reciprocity that

s_{∗}(I^{n}(K)) = s_{∗}(I^{n−1}(F)I(K)) = I^{n−1}(F)s_{∗}(I(K)) ⊂ I^{n−1}(F)I(F) = I^{n}(F).

(2): This follows from (1) if char F ≠ 2 and from Lemma 34.16(2) and Frobenius Reciprocity if char F = 2. □

Lemma 34.18. Let K/F be a quadratic extension and s, s′ : K → F two nonzero F -linear functionals. Let 𝔟 ∈ I^{n}(K). Then s_{∗}(𝔟) ≡ s′_{∗}(𝔟) mod I^{n+1}(F).

Proof. As in the proof of Corollary 20.7, there exists a c ∈ K^{×} such that s′_{∗}(𝔠) = s_{∗}(c𝔠) for all symmetric bilinear forms 𝔠. As 𝔟 ∈ I^{n}(K), we have ⟨⟨c⟩⟩ · 𝔟 ∈ I^{n+1}(K). Consequently, s_{∗}(𝔟) − s′_{∗}(𝔟) = s_{∗}(⟨⟨c⟩⟩ · 𝔟) lies in I^{n+1}(F). The result follows. □

Corollary 34.19. Let K/F be a quadratic field extension and s : K → F a nontrivial F -linear functional. Then s_{∗}(⟨⟨x⟩⟩) ≡ ⟨⟨N_{K/F} (x)⟩⟩ modulo I^{2}(F) for every x ∈ K^{×}.

Proof. By Lemma 34.18, we know that s_{∗}(⟨⟨x⟩⟩) is independent of the nontrivial F -linear functional s modulo I^{2}(F). Using the functional defined in (20.8), the result follows by Corollary 20.13. □

Let K/F be a separable quadratic field extension and let s : K → F be a nontrivial F -linear functional such that s(1) = 0. It follows from Theorem 34.9 and Corollary 34.17 that we have a well-defined complex (34.20)

I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} I^{n}(F) ⟶^{· N_{K/F}} I_{q}^{n+1}(F) ⟶^{r_{K/F}} I_{q}^{n+1}(K) ⟶^{s_{∗}} I_{q}^{n+1}(F)

and this induces (where by, abuse of notation, we label the maps in the same way) (34.21)

\overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{· N_{K/F}} \overline{I}_{q}^{n+1}(F) ⟶^{r_{K/F}} \overline{I}_{q}^{n+1}(K) ⟶^{s_{∗}} \overline{I}_{q}^{n+1}(F).

where \overline{I}^{n}(F) := I^{n}(F)/I^{n+1}(F).

By Lemma 34.18, it follows that the homomorphism s_{∗} in (34.21) is independent of the nontrivial F -linear functional K → F although it is not independent in (34.20).

We show that the complexes (34.20) and (34.21) are exact on bilinear Pfister forms. More precisely, we have

Theorem 34.22. Let K/F be a separable quadratic field extension and s : K → F a nontrivial F -linear functional such that s(1) = 0.

(1) Let 𝔠 be an anisotropic bilinear n-fold Pfister form over K. If s_{∗}(𝔠) ∈ I^{n+1}(F), then there exists a bilinear n-fold Pfister form 𝔟 over F such that 𝔠 ≃ 𝔟_{K}.

(2) Let 𝔟 be an anisotropic bilinear n-fold Pfister form over F. If 𝔟 · N_{K/F} ∈ I^{n+2}(F), then there exists a bilinear n-fold Pfister form 𝔠 over K such that 𝔟 = s_{∗}(𝔠).

(3) Let φ be an anisotropic quadratic (n + 1)-fold Pfister form over F. If r_{K/F} (φ) ∈ I^{n+2}(K), then there exists a bilinear n-fold Pfister form 𝔟 over F such that φ ≃ 𝔟 ⊗ N_{K/F}.

(4) Let ψ be an anisotropic (n + 1)-fold quadratic Pfister form over K. If s_{∗}(ψ) ∈ I^{n+2}(F), then there exists a quadratic (n + 1)-fold Pfister form φ over F such that ψ ≃ φ_{K}.

Proof. (1): As 𝔠 represents 1, the form s_{∗}(𝔠) is isotropic and belongs to I^{n+1}(F). It follows from the Hauptsatz 23.7 that s_{∗}(𝔠) = 0 in W (F). We show by induction on k ≥ 0 that there is a bilinear k-fold Pfister form 𝔡 over F and a bilinear (n − k)-fold Pfister form 𝔢 over K such that 𝔠 ≃ 𝔡_{K} ⊗ 𝔢. The statement that we need follows when k = n.

Suppose we have 𝔡 and 𝔢 for some k < n. We have

0 = s_{∗}(𝔠) = s_{∗}(𝔡_{K} · 𝔢′ ⊥ 𝔡_{K}) = s_{∗}(𝔡_{K} · 𝔢′)

in W (F) where, as usual, 𝔢′ denotes the pure subform of 𝔢. In particular, s_{∗}(𝔡_{K} ⊗𝔢′) is isotropic. Thus there exists b ∈ F^{×} ∩ D(𝔡_{K} ⊗ 𝔢′). It follows that 𝔡_{K} ⊗ 𝔢 ≃ 𝔡_{K} ⊗ ⟨⟨b⟩⟩ ⊗ 𝔣 for some Pfister form 𝔣 over K by Theorem 6.15.

(2): By the Hauptsatz 23.7, we have 𝔟 ⊗ N_{K/F} is hyperbolic. We claim that 𝔟 ≃ ⟨⟨a⟩⟩⊗𝔞 for some a ∈ N_{K/F} (K^{×}) and an (n−1)-fold bilinear Pfister form 𝔞 over F. If char F ≠ 2, the claim follows from Corollary 6.14. If char F = 2, it follows from Lemma 9.11 that N_{K/F} ≃ ⟨⟨a]] for some a ∈ D(𝔟′). Clearly, a ∈ N_{K/F} (K^{×}) and by Lemma 6.11, the form 𝔟 is divisible by ⟨⟨a⟩⟩. The claim is proven.

As a ∈ N_{K/F} (K^{×}) there is y ∈ K^{×} such that s_{∗}(⟨⟨y⟩⟩) = ⟨⟨a⟩⟩. It follows that s_{∗}(⟨⟨y⟩⟩ · 𝔞) = ⟨⟨a⟩⟩ · 𝔞 = 𝔟.

(3): By the Hauptsatz 23.7, we have r_{K/F} (φ) = 0 in I_{q}(K). The field K is isomorphic to the function field of the 1-fold Pfister form N_{K/F}. The statement now follows from Corollary 23.6.

(4): In the case char F ≠ 2, the statement follows from (1). So we may assume that char F = 2. As ψ represents 1, the form s_{∗}(ψ) is isotropic and belongs to I_{q}^{n+2}(F). It follows from the Hauptsatz 23.7 that s_{∗}(ψ) = 0 ∈ I_{q}(F). We show by induction for each k ∈ [1, n] that there is a k-fold bilinear Pfister form 𝔡 over F and a quadratic Pfister form ρ over K such that ψ ≃ 𝔡_{K} ⊗ ρ.

Suppose we have 𝔡 and ρ for some k < n. As dim(𝔡_{K} ⊗ρ′) > \frac{1}{2} dim(𝔡_{K} ⊗ρ) with ρ′ the pure subform of ρ, the subspace of s_{∗}(𝔡_{K} ⊗ ρ′) intersects a totally isotropic subspace of s_{∗}(𝔡_{K} ⊗ ρ) and therefore is isotropic. Hence there is a c ∈ F satisfying c ∈ D(𝔡_{K} ⊗ ρ) \ D(𝔡_{K}). By Proposition 15.7, ψ ≃ 𝔡 ⊗ ⟨⟨c⟩⟩_{K} ⊗ μ for some quadratic Pfister form μ.

Applying the statement with k = n, we get an n-fold bilinear Pfister form 𝔟 over F such that ψ ≃ 𝔟_{K}⊗⟨⟨y]] for some y ∈ K. As s_{∗}(⟨⟨y]]) is similar to ⟨⟨Tr_{K/F} (y)]], we have 𝔟 ⊗ ⟨⟨Tr_{K/F} (y)]] = 0 ∈ I_{q}(F). By Corollary 6.14, Tr_{K/F} (y) = b + b^{2} + 𝔟′(v, v) for some b ∈ F and v ∈ V_{𝔟′}. Let x ∈ K \ F be an element such that x^{2} + x + a = 0 for some a ∈ F. Set z = xb + (xb)^{2} + 𝔟′_{K}(xv, xv) ∈ K and c = y + z. Since Tr_{K/F} (x) = Tr_{K/F} (x^{2}) = 1, we have Tr_{K/F} (z) = Tr_{K/F} (y). It follows that c ∈ F.

Again by Corollary 6.14, we see that 𝔟_{K} ⊗ ⟨⟨z]] is hyperbolic and therefore

ψ = 𝔟_{K} · ⟨⟨y]] = 𝔟_{K} · ⟨⟨y + z]] = (𝔟 · ⟨⟨c]])_{K}.  □

Remark 34.23. Suppose that char F ≠ 2 and K = F (\sqrt{a}) is a quadratic extension of F. Let 𝔟 be an anisotropic bilinear n-fold Pfister form over F. Then N_{K/F} = ⟨⟨a⟩⟩, so by Theorem 34.22(3), the following are equivalent:

(1) 𝔟_{K} ∈ I^{n+1}(K).

(2) 𝔟 ∈ ⟨⟨a⟩⟩W (F).

(3) 𝔟 ≃ ⟨⟨a⟩⟩ ⊗ 𝔠 for some (n − 1)-fold Pfister form 𝔠.

We now consider the case of a purely inseparable quadratic field extension K/F.

Lemma 34.24. Let K/F be a purely inseparable quadratic field extension and s : K → F a nonzero F -linear functional satisfying s(1) = 0. Let b ∈ F^{×}. Then the following conditions are equivalent:

(1) b ∈ N_{K/F} (K^{×}).

(2) ⟨⟨b⟩⟩_{K} = 0 ∈ W (K).
(3) ⟨⟨b⟩⟩ = s_{∗}(⟨y⟩) for some y ∈ K^{×}.

Proof. The equality N_{K/F} (K^{×}) = K^{2} ∩ F^{×} proves (1) ⇔ (2). For any y ∈ F^{×}, it follows by Corollary 34.19 that s_{∗}(⟨y⟩) is similar to ⟨⟨N_{K/F} (y)⟩⟩. This proves that (1) ⇔ (3). □

Proposition 34.25. Let K/F be a purely inseparable quadratic field extension and s : K → F a nontrivial F -linear functional such that s(1) = 0. Let 𝔟 be an anisotropic bilinear form over F. Then there exist bilinear forms 𝔠 over K and 𝔡 over F satisfying 𝔟 ≃ s_{∗}(𝔠) ⊥ 𝔡 and 𝔡_{K} is anisotropic.

Proof. We induct on dim 𝔟. Suppose that 𝔟_{K} is isotropic. Then there is a 2-dimensional subspace W ⊂ V_{𝔟} such that (𝔟|_{W})_{K} is isotropic. By Lemma 34.24, we have 𝔟|_{W} ≃ s_{∗}(⟨y⟩) for some y ∈ K^{×}. Applying the induction hypothesis to the orthogonal complement of W in V completes the proof. □

Theorem 34.4 and Proposition 34.25 yield

Corollary 34.26. Let K/F be a purely inseparable quadratic field extension and s : K → F a nonzero F -linear functional such that s(1) = 0. Then the sequence

W (F) ⟶^{r_{K/F}} W (K) ⟶^{s_{∗}} W (F) ⟶^{r_{K/F}} W (K)

is exact.

Let K/F be a purely inseparable quadratic field extension and s : K → F a nonzero linear functional such that s(1) = 0. It follows from Corollaries 34.17 and 34.26 that we have well-defined complexes

(34.27)  I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} I^{n}(F) ⟶^{r_{K/F}} I^{n}(K)

and

(34.28)  \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K).

As in the separable case, the homomorphism s_{∗} in (34.28) is independent of the nontrivial F -linear functional K → F by Lemma 34.18 although it is not independent in (34.27).

We show that the complexes (34.27) and (34.28) are exact on quadratic Pfister forms.

Theorem 34.29. Let K/F be a purely inseparable quadratic field extension and s : K → F a nontrivial F -linear functional such that s(1) = 0.

(1) Let 𝔠 be anisotropic n-fold bilinear Pfister form over K. If s_{∗}(𝔠) ∈ I^{n+1}(F), then there exists an 𝔟 over K such that 𝔠 ≃ 𝔟_{K}.

(2) Let 𝔟 be anisotropic n-fold bilinear Pfister form over F. If 𝔟_{K} ∈ I^{n+1}(K), then there exists an n-fold bilinear Pfister form 𝔠 such that 𝔟 = s_{∗}(𝔠).

Proof. (1): The proof is the same as in Theorem 34.22(1).

(2): By the Hauptsatz 23.7, we have 𝔟_{K} = 0 ∈ W (K). In particular, 𝔟_{K} is isotropic and hence there is a 2-dimensional subspace W ⊂ V_{𝔟} with 𝔟|_{W} nondegenerate and isotropic over K. Hence 𝔟|_{W} is similar to ⟨⟨b⟩⟩ for some b ∈ F^{×}. As ⟨⟨b⟩⟩_{K} = 0, by Lemma 34.24 ⟨⟨b⟩⟩ = s_{∗}(⟨⟨y⟩⟩) for some y ∈ K^{×}. By Corollary 6.17, we have 𝔟 ≃ ⟨⟨b⟩⟩ ⊗ 𝔡 for some bilinear Pfister form 𝔡. Finally,

𝔟 = ⟨⟨b⟩⟩ · 𝔡 = s_{∗}(⟨⟨y⟩⟩) · 𝔡 = s_{∗}(⟨⟨y⟩⟩ · 𝔡) ∈ W (F).  □

We shall show in Theorems 40.3, 40.5, and 40.6 that the complexes (34.20), (34.21), (34.27) and (34.28) are exact for any n. Note that the exactness for small n (up to 2) can be shown by elementary means.

34.D. Torsion in the Witt ring under a quadratic extension. We turn to the transfer of the torsion ideal in the Witt ring of a quadratic extension to obtain results found in [31]. We need the following lemma.

Lemma 34.30. Let K/F be a quadratic field extension of F and 𝔟 a bilinear Pfister form over F.

(1) If 𝔠 is an anisotropic bilinear form over K such that 𝔟_{K} ⊗𝔠 is defined over F, then there exists a form 𝔡 over F such that 𝔟_{K} ⊗ 𝔠 ≃ (𝔟 ⊗ 𝔡)_{K}.

(2) r_{K/F} (W (F)) ∩ 𝔟_{K}W (K) = r_{K/F} (𝔟W (F)).

Proof. (1): Let 𝔠 = ⟨a_{1}, …, a_{n}⟩. We induct on dim 𝔠 = n. By hypothesis, there is a c ∈ F^{×} ∩ D(𝔟_{K} ⊗ 𝔠). Write c = a_{1}b_{1} + ⋯ + a_{n}b_{n} with b_{i} ∈ D̃(𝔟_{K}). Let c_{i} = b_{i} if b_{i} ≠ 0 and 1 if not. Then 𝔢 := ⟨a_{1}c_{1}, …, a_{n}c_{n}⟩ represents c so 𝔢 ≃ ⟨c⟩ ⊥ 𝔣. Since b_{i} ∈ G_{K}(𝔟), we have

𝔟_{K} ⊗ 𝔠 ≃ 𝔟_{K} ⊗ 𝔢 ≃ 𝔟_{K} ⊗ ⟨c⟩ ⊥ 𝔟_{K} ⊗ 𝔣.

As 𝔟_{K} ⊗ 𝔣 ∈ im(r_{K/F}), its anisotropic part is defined over F by Proposition 34.1 and Theorem 34.4. By induction, there exists a form 𝔤 such that 𝔟_{K} ⊗𝔣 ≃ 𝔟_{K} ⊗𝔤_{K}. Then ⟨c⟩ ⊥ 𝔤 works.

(2) follows easily from (1). □

Proposition 34.31. Let K = F (\sqrt{a})/F be a quadratic extension with a ∈ F^{×} and s : K → F a nontrivial F -linear functional such that s(1) = 0. Let 𝔟 be an n-fold bilinear Pfister form. Then

s_{∗}(W (K)) ∩ ann_{W (F)}(𝔟) = s_{∗}(ann_{W (K)}(𝔟_{K})).

Proof. By Frobenius Reciprocity, we have

s_{∗}(ann_{W (K)}(𝔟_{K})) ⊂ s_{∗}(W (K)) ∩ ann_{W (F)}(𝔟).

35. TORSION IN I^{n}(F) AND TORSION PFISTER FORMS 147

Conversely, if 𝔠 ∈ s_{∗}(W (K)) ∩ ann_{W (F)}(𝔟), we can write 𝔠 = s_{∗}(𝔡) for some form 𝔡 over K. By Theorem 34.4 and Lemma 34.30,

𝔟_{K} ⊗ 𝔡 ∈ r_{K/F} (W (F)) ∩ 𝔟_{K} W (K) = r_{K/F} (𝔟W (F)).

Hence there exists a form 𝔢 over F such that 𝔟_{K} ⊗ 𝔡 = (𝔟 ⊗ 𝔢)_{K}. Let 𝔣 = 𝔡 ⊥ −𝔢_{K}. Then 𝔠 = s_{∗}(𝔡) = s_{∗}(𝔣) ∈ s_{∗}(ann_{W (K)}(𝔟_{K})) as needed. □

The torsion W_{t}(F) of W (F) is 2-primary. Thus applying the proposition to ρ = 2^{n}⟨1⟩ for all n yields

Corollary 34.32. Let K = F (\sqrt{a}) be a quadratic extension of F with a ∈ F^{×} and s : K → F a nontrivial F -linear functional such that s(1) = 0. Then W_{t}(F) ∩ s_{∗}(W (K)) = s_{∗}(W_{t}(K)).

We also have the following:

Corollary 34.33. Suppose that F is a field of characteristic different from 2 and K = F (\sqrt{a}) a quadratic extension of F. Let s : K → F be a nontrivial F -linear functional such that s(1) = 0. Then

⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(2⟨1⟩) = Ker(r_{K/F}) ∩ s_{∗}(W (K))
⊂ ann_{W (F)}(2⟨1⟩) ∩ ann_{W (F)}(⟨⟨a⟩⟩) = s_{∗}(ann_{W (K)}(2⟨1⟩)).
Proof. As ⟨⟨a, a⟩⟩ ≃ ⟨⟨−1, a⟩⟩, we have
⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(2⟨1⟩) = ⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(⟨⟨a⟩⟩),

which yields the first equality by Corollary 34.12. As ⟨⟨a⟩⟩W (F) ⊂ ann_{W (F)}(⟨⟨a⟩⟩), we have the inclusion. Finally, s_{∗}(W (K)) ∩ ann_{W (F)}(2⟨1⟩) = s_{∗}(ann_{W (K)}(2⟨1⟩_{K})) by Proposition 34.31, so Corollary 34.12 yields the second equality. □

Remark 34.34. Suppose that F is a formally real field and K a quadratic extension. Let s_{∗} : W (K) → W (F) be the transfer induced by a nontrivial F -linear functional such that s(1) = 0. Then it follows by Corollaries 34.12 and 34.32 that the maps induced by r_{K/F} and s_{∗} yield an exact sequence

0 → W_{red}(K/F) → W_{red}(F) ⟶^{r_{K/F}} W_{red}(K) ⟶^{s_{∗}} W_{red}(F)

(again abusing notation for the maps) where

W_{red}(K/F) := Ker(W_{red}(F) → W_{red}(K)).

By Corollary 33.15, we have a zero sequence

0 → I_{red}^{n}(K/F) → I_{red}^{n}(F) ⟶^{r_{K/F}} I_{red}^{n}(K) ⟶^{s_{∗}} I_{red}^{n}(F)

where I_{red}^{n}(K/F) := Ker(I_{red}^{n}(F) → I_{red}^{n}(K)).

In fact, we shall see in §41 that this sequence is also exact.

35. Torsion in I^{n}(F) and torsion Pfister forms

In this section we study the property that I(F) is nilpotent, i.e., that there exists an n such that I^{n}(F) = 0. For such an n to exist, the field must be nonformally real. In order to study all fields we broaden this investigation to the study of the existence of an n such that I^{n}(F) is torsion-free. We wish to establish the relationship between this occurring over F and over a quadratic field extension K. This more general case is more difficult, so in this section we look at the simpler property that there are no torsion bilinear n-fold Pfister forms over the field F. This was the approach introduced and used in [37] and would be equivalent to I^{n}(F) being torsion-free if we knew that torsion bilinear n-fold Pfister forms generate the torsion in I^{n}(F). This is in fact true as shown in [8] and which we shall show in §41, but cannot be proven by these elementary methods.

35.A. The property A_{n}. In this section we begin to study torsion in I^{n}(F) for a field F. We set

I_{t}^{n}(F) := W_{t}(F) ∩ I^{n}(F).

Note that the group I_{t}^{1}(F) = I_{t}(F) is generated by torsion binary forms by Proposition 31.27.

It is obvious that

I_{t}^{n}(F) ⊃ I^{n−1}(F)I_{t}(F).

Proposition 35.1. I_{t}^{2}(F) = I(F)I_{t}(F).

Proof. Note that for all a, a′ ∈ F^{×} and w, w′ ∈ D(∞⟨1⟩), we have
a⟨⟨w⟩⟩ + a′⟨⟨w′⟩⟩ = a⟨⟨−aa′, w⟩⟩ + a′w⟨⟨ww′⟩⟩,

hence

a⟨⟨w⟩⟩ + a′⟨⟨w′⟩⟩ ≡ a′w⟨⟨ww′⟩⟩ mod I(F)I_{t}(F).

Let 𝔟 ∈ I_{t}^{2}(F). By Proposition 31.27, we have 𝔟 is a sum of binary forms a⟨⟨w⟩⟩ with a ∈ F^{×} and w ∈ D(∞⟨1⟩). Repeated application of the congruence above shows that 𝔟 is congruent to a binary form a⟨⟨w⟩⟩ modulo I(F)I_{t}(F). As a⟨⟨w⟩⟩ ∈ I^{2}(F) we have a⟨⟨w⟩⟩ = 0 and therefore 𝔟 ∈ I(F)I_{t}(F). □

We shall prove in §41 that the equality I_{t}^{n}(F) = I^{n−1}(F)I_{t}(F) holds for every n.

It is easy to determine Pfister forms of order 2 (cf. Corollary 6.14).

Lemma 35.2. Let 𝔟 be a bilinear n-fold Pfister form. Then 2𝔟 = 0 in W (F) if and only if either char F = 2 or 𝔟 = ⟨⟨w⟩⟩ ⊗ 𝔠 for some w ∈ D(2⟨1⟩) and 𝔠 is an (n − 1)-fold Pfister form.

Proposition 35.3. Let F be a field and n ≥ 1 an integer. The following conditions are equivalent:

(1) There are no n-fold Pfister forms of order 2 in W (F).

(2) There are no anisotropic n-fold Pfister forms of finite order in W (F).

(3) For every m ≥ n, there are no anisotropic m-fold Pfister forms of finite order in W (F).

Proof. The implications (3) ⇒ (2) ⇒ (1) are trivial.

(1) ⇒ (3): If char F = 2 the statement is clear as W (F) is torsion. Assume that char F ≠ 2. Let 2^{k}𝔟 = 0 in W (F) for some k ≥ 1 and 𝔟 an m-fold Pfister form with m ≥ n. We show by induction on k that 𝔟 = 0 in W (F). It follows from Lemma 35.2 that 2^{k−1}𝔟 ≃ ⟨⟨w⟩⟩ ⊗ 𝔠 for some w ∈ D(2⟨1⟩) and a (k + m − 2)-fold Pfister form 𝔠. Let 𝔡 be an (n − 1)-fold Pfister form dividing 𝔠. Again by Lemma 35.2, the form 2⟨⟨w⟩⟩ · 𝔡 is 0 in W (F), hence by assumption, ⟨⟨w⟩⟩ · 𝔡 = 0 in W (F). It follows that 2^{k−1}𝔟 = ⟨⟨w⟩⟩ · 𝔠 = 0 in W (F). By the induction hypothesis, 𝔟 = 0 in W (F). □

We say that a field F satisfies A_{n} if the equivalent conditions of Proposition 35.3 hold. It follows from the definition that the condition A_{n} implies A_{m} for every m ≥ n. It follows from Proposition 31.9 that F satisfies A_{1} if and only if F is pythagorean.

If F is not formally real, the condition A_{n} is equivalent to I^{n}(F) = 0 as the group W (F) is torsion.

As the group I_{t}(F) is generated by torsion binary forms, the property A_{n} implies that I^{n−1}(F)I_{t}(F) = 0.

Exercise 35.4. Suppose that F is a field of characteristic not 2. If K is a quadratic extension of F, let s^{K} : K → F be a nontrivial F -linear functional such that s^{K}(1) = 0. Show the following are equivalent:

(1) F satisfies A_{n+1}.

(2) s_{∗}^{F (\sqrt{w})}(P_{n}(F (\sqrt{w}))) = P_{n}(F) for every w ∈ D(∞⟨1⟩).

(3) s_{∗}^{F (\sqrt{w})}(I^{n}(F (\sqrt{w}))) = I^{n}(F) for every w ∈ D(∞⟨1⟩).

We now study the property A_{n} under field extensions. The case of fields of characteristic 2 is easy.

Lemma 35.5. Let K/F be a finite extension of fields of characteristic 2. Then I^{n}(F) = 0 if and only if I^{n}(K) = 0.

Proof. The property I^{n}(E) = 0 for a field E is equivalent to [E : E^{2}] < 2^{n} by Example 6.5. We have [K : F] = [K^{2} : F^{2}], as the Frobenius map K → K^{2} given by x → x^{2} is an isomorphism. Hence

(35.6)  [K : K^{2}] = [K : F^{2}]/[K^{2} : F^{2}] = [K : F^{2}]/[K : F] = [F : F^{2}].

Thus we have I^{n}(K) = 0 if and only if I^{n}(F) = 0. □

Let F_{0} be a formally real field satisfying A_{1}, i.e., a pythagorean field. Let F_{n} = F_{0}((t_{1})) ⋯ ((t_{n})) be the iterated Laurent series field over F_{0}. Then F_{n} is also formally real pythagorean (cf. Example 31.6), hence F_{n} satisfies A_{n} for all n ≥ 1. However, K_{n} = F_{n}(\sqrt{−1}) does not satisfy A_{n} as ⟨⟨t_{1}, …, t_{n}⟩⟩ is an anisotropic form over the nonformally real field K_{n}. Thus the property A_{n} is not preserved under quadratic extensions. Nevertheless, we have

Proposition 35.7. Suppose that F satisfies A_{n}. Let K = F (\sqrt{a}) be a quadratic extension of F with a ∈ F^{×}. Then K satisfies A_{n} if either of the following two conditions hold:

(1) a ∈ D(∞⟨1⟩).

(2) Every bilinear n-fold Pfister form over F becomes metabolic over K.

Proof. If char F = 2, then I^{n}(F) = 0, hence I^{n}(K) = 0 by Lemma 35.5. So we may assume that char F ≠ 2. Let y ∈ K^{×} satisfy y ∈ D(2⟨1⟩_{K}) and let 𝔢 be an (n − 1)-fold Pfister form over K. By Lemma 35.2, it suffices to show that 𝔟 := ⟨⟨y⟩⟩ ⊗ 𝔢 is trivial in W (K). Let s_{∗} : W (K) → W (F) be the transfer induced by a nontrivial F -linear functional s(1) = 0.

We claim that s_{∗}(𝔟) = 0. Suppose that n = 1. Then s_{∗}(𝔟) ∈ I_{t}(F) = 0. So we may assume that n ≥ 2. As I^{n−1}(K) is generated by Pfister forms of the form ⟨⟨z⟩⟩ ⊗ 𝔡_{K} with z ∈ K^{×} and 𝔡 an (n − 2)-fold Pfister form over F by Lemma 34.16, we may assume that 𝔟 = ⟨⟨y, z⟩⟩ ⊗ 𝔡_{K}.

We have s_{∗}(⟨⟨y, z⟩⟩) ∈ I_{t}^{2}(F) = I(F)I_{t}(F) by Proposition 35.1. So
s_{∗}(⟨⟨y, z⟩⟩ · 𝔡_{K}) = s_{∗}(⟨⟨y, z⟩⟩) · 𝔡

lies in I^{n−1}(F)I_{t}(F) which is trivial by A_{n}. The claim is proven.

It follows that 𝔟 = 𝔠_{K} for some n-fold Pfister form 𝔠 over F by Theorem 34.22. Thus we are done if every n-fold Pfister form over F becomes hyperbolic over K. So assume that a ∈ D(∞⟨1⟩). As 𝔟 is torsion in W (K), there exists an m such that 2^{m}𝔟 = 0 in W (F). Thus 2^{m}𝔠_{K} is hyperbolic so 2^{m}𝔠 is a sum of binary forms x⟨⟨ay^{2} + x^{2}⟩⟩ in W (F) for some x, y, z in F by Corollary 34.12. In particular, 2^{m}𝔠 is torsion and so trivial by A_{n} for F. The result follows. □

Corollary 35.8. Suppose that I^{n}(F) = 0 (in particular, F is not formally real). Let K/F be a quadratic extension. Then I^{n}(K) = 0.

In general, the above corollary does not hold if K/F is not quadratic. For example, let F be the quadratic closure of the rationals, so I(F) = 0. There exist algebraic extensions K of F such that I(K) ≠ 0, e.g., K = F (\sqrt[3]{2}). It is true, however, that in this case I^{2}(K) = 0. It is still an unanswered question whether I^{2}(K) = 0 when K/F is finite and F is an arbitrary quadratically closed field, equivalently whether the cohomological 2-dimension of a quadratically closed field is at most one.

If I^{n}(F) is torsion-free, then F satisfies A_{n}. Conversely, if F satisfies A_{1}, then I(F) is torsion-free by Proposition 31.9. If F satisfies A_{2}, then it follows from Proposition 35.1 that I^{2}(F) is torsion-free as I_{t}(F) is generated by torsion binary forms.

Proposition 35.9. A field F satisfies A_{3} if and only if I^{3}(F) is torsion-free.

Proof. The statement is obvious if F is not formally real, so we may assume that char F ≠ 2. Let 𝔟 ∈ I^{3}(F) be a torsion element. By Proposition 35.1,

𝔟 = ∑_{i=1}^{r} x_{i}⟨⟨y_{i}, w_{i}⟩⟩

for some x_{i}, y_{i} ∈ F^{×} and w_{i} ∈ D(∞⟨1⟩). We show by induction on r that 𝔟 = 0.

It follows from Proposition 35.7 that K = F (\sqrt{w}) with w = w_{r} satisfies A_{3}. By the induction hypothesis, we have 𝔟_{K} = 0. Thus 𝔟 = ⟨⟨w⟩⟩ · 𝔠 for some 𝔠 ∈ W (F) by Corollary 34.12. Then 𝔠 must be even-dimensional as the determinant of 𝔠 is trivial. Choose d ∈ F^{×} such that 𝔡 := 𝔠 + ⟨⟨d⟩⟩ ∈ I^{2}(F).

Thus in W (F),

𝔟 = ⟨⟨w⟩⟩ · 𝔡 − ⟨⟨w, d⟩⟩.

Note that ⟨⟨w⟩⟩ · 𝔡 = 0 in W (F) by A_{3}. Consequently, ⟨⟨w, d⟩⟩ ∈ I^{3}(F), so it is zero in W (F) by the Hauptsatz 23.7. This shows 𝔟 = 0. □

We shall show in Corollary 41.5 below that I^{n}(F) is torsion-free if and only if F satisfies A_{n} for every n ≥ 1.

We have an application for quadratic forms first shown in [36].

Theorem 35.10 (Classification Theorem). Let F be a field.

(1) Dimension and total signature classify the isometry classes of nondegenerate quadratic forms over F if and only if I_{q}(F) is torsion-free, i.e., F is pythagorean. In particular, if F is not formally real, then dimension classifies the isometry classes of forms over F if and only if F is quadratically closed.

(2) Dimension, discriminant and total signature classify the isometry classes of nondegenerate even-dimensional quadratic forms over F if and only if I_{q}^{2}(F) is torsion-free. In particular, if F is not formally real, then dimension and discriminant classify the isometry classes of forms over F if and only if I_{q}^{2}(F) = 0.

(3) Dimension, discriminant, Clifford invariant, and total signature classify the isometry classes of nondegenerate even-dimensional quadratic forms over F if and only if I_{q}^{3}(F) is torsion-free. In particular, if F is not formally real, then dimension, discriminant, and Clifford invariant classify the isometry classes of forms over F if and only if I_{q}^{3}(F) = 0.

Proof. We prove (3) as the others are similar (and easier). If I_{q}^{3}(F) is not torsion-free, then there exists an anisotropic torsion form φ ∈ P_{3}(F) by Proposition 35.9 if F is formally real and trivially if F is not formally real as then I_{q}(F) is torsion. As φ and 4ℍ have the same dimension, discriminant, Clifford invariant, and total signature but are not isometric, these invariants do not classify.

Conversely, assume that I_{q}^{3}(F) is torsion-free. Let φ and ψ be nondegenerate even-dimensional quadratic forms having the same dimension, discriminant, Clifford invariant, and total signature. Then by Theorem 13.7, the form θ := φ ⊥ −ψ lies in I_{q}^{2}(F) and is torsion. As φ and ψ have the same dimension, it suffices to show that θ is hyperbolic. Thus the result is equivalent to showing that if a torsion form θ ∈ I_{q}^{2}(F) has trivial Clifford invariant and I_{q}^{3}(F) is torsion-free, then θ is hyperbolic.

The case char F = 2 follows from Theorem 16.3. So we may assume that char F ≠ 2. By Proposition 35.1, we can write θ = ∑_{i=1}^{r} a_{i}⟨⟨b_{i}, c_{i}⟩⟩ in I_{q}(F) with ⟨⟨c_{i}⟩⟩ torsion forms. We prove that θ is hyperbolic by induction on r.

Let K = F (\sqrt{c}) with c = c_{r}. Clearly, θ_{K} ∈ I_{q}^{2}(K) is torsion and has trivial Clifford invariant. By Proposition 35.7 and Corollary 35.9, we have I_{q}^{3}(K) is torsionfree. By the induction hypothesis, θ_{K} is hyperbolic. By Corollary 23.6, we conclude that θ = ψ · ⟨⟨c⟩⟩ in I_{q}(F) for some quadratic form ψ. As disc(θ) is trivial, dim ψ is even. Choose d ∈ F^{×} such that τ := ψ + ⟨⟨d⟩⟩ ∈ I^{2}(F). Then

θ = τ · ⟨⟨c⟩⟩ − ⟨⟨d, c⟩⟩

in W (F).

As the torsion form τ ⊗ ⟨⟨c⟩⟩ belongs to I_{q}^{3}(F), it is hyperbolic. As the Clifford invariant of θ is trivial, it follows that the Clifford invariant of ⟨⟨d, c⟩⟩ must also be trivial. By Corollary 12.5, ⟨⟨d, c⟩⟩ is hyperbolic and hence θ is hyperbolic. □

Remark 35.11. The Stiefel-Whitney classes introduced in (5.4) are defined on nondegenerate bilinear forms. If 𝔟 is such a form then the w_{i}(𝔟) determine sgn 𝔟 for every P ∈ 𝔛(F) by Remark 5.8 and Example 5.13. We also have w_{i} = e_{i} for i = 1, 2 by Corollary 5.9.

Let 𝔟 and 𝔟′ be two nondegenerate symmetric bilinear forms of the same dimension. Suppose that w(𝔟) = w(𝔟′), then w([𝔟] − [𝔟′]) = 1, where [ ] is the class of a form in the Witt-Grothendieck ring Ŵ(F). It follows that [𝔟] − [𝔟′] lies in Î^{3}(F) by (5.11), hence 𝔟 − 𝔟′ lies in I^{3}(F). As the w_{i} determine the total signature of a form, we have 𝔟 − 𝔟′ is torsion by the Local-Global Principle 31.22. It follows that the dimension and total Stiefel-Whitney class determine the isometry class of anisotropic bilinear forms if and only if I^{3}(F) is torsion-free.

Suppose that char F ≠ 2. Then all metabolic forms are hyperbolic, so in this case the dimension and total Stiefel-Whitney class determine the isometry class of nondegenerate symmetric bilinear forms if and only if I^{3}(F) is torsion-free. In addition, we can define another Stiefel-Whitney map

ŵ : Ŵ(F) → H^{∗}(F)[[t]]^{×}

to be the composition of w and the map k_{∗}(F)[[t]] → H^{∗}(F)[[t]] induced by the norm residue homomorphism h_{F}^{∗} : k_{∗}(F) → H^{∗}(F) in §101.5. Then dimension and ŵ classify the isometry classes of nondegenerate bilinear forms if and only if I^{3}(F) is torsion-free by Theorem 35.10 as h_{∗} is an isomorphism if F is a real closed field and w̃_{2} is the classical Hasse invariant so determines the Clifford invariant.

We turn to the question on whether the property A_{n} goes down. This was first done in [37]. We use the proof given in [31].

Theorem 35.12. Let K/F be a finite normal extension. If K satisfies A_{n} so does F.

Proof. Let G = Gal(K/F) and let H be a Sylow 2-subgroup of G. Set E = K^{H}, L = K^{G}. The field extension L/F is purely inseparable, so [L : F] is either odd or L/F is a tower of successive quadratic extensions. The extension K/E is a tower of successive quadratic extensions and [E : L] is odd. Thus we may assume that [K : F] is either 2 or odd. Springer’s Theorem 18.5 solves the case of odd degree. Hence we may assume that K/F is a quadratic extension.

The case char F = 2 follows from Lemma 35.5. Thus we may assume that the characteristic of F is different from 2 and therefore K = F (\sqrt{a}) with a ∈ F^{×}. Let s : K → F be a nontrivial F -linear functional with s(1) = 0.

Let 𝔟 be a 2-torsion bilinear n-fold Pfister form. We must show that 𝔟 = 0 in W (F). As 𝔟_{K} = 0 we have

𝔟 ∈ ⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(2⟨1⟩)

by Corollary 34.12. As ⟨⟨a, a⟩⟩ = ⟨⟨a, −1⟩⟩, it follows that ⟨⟨a⟩⟩ · 𝔟 = 0 in W (F), hence by Corollary 6.14, we can write 𝔟 ≃ ⟨⟨b⟩⟩ ⊗ 𝔠 for some (n − 1)-fold Pfister form 𝔠 and b ∈ D(⟨⟨a⟩⟩). Choose x ∈ K^{×} such that s_{∗}(⟨x⟩) = ⟨⟨b⟩⟩ and let 𝔡 = x𝔠_{K}. Then

s_{∗}(𝔡) = s_{∗}(⟨x⟩)𝔠 = ⟨⟨b⟩⟩𝔠 = 𝔟.

If 𝔡 = 0, then 𝔟 = 0 and we are done. So we may assume that 𝔡 and therefore 𝔠_{K} is anisotropic.

We have s_{∗}(2𝔡) = 2𝔟 = 0 in W (F), hence the form s_{∗}(2𝔡) is isotropic. Therefore, 2𝔡 represents an element c ∈ F^{×}, hence there exist u, v ∈ D̃(𝔠_{K}) satisfying x(u+v) = c. But the form ⟨⟨u+v⟩⟩⊗𝔠 is 2-torsion and K satisfies A_{n}. Consequently, u + v ∈ D(𝔠_{K}) = G(𝔠_{K}) as 𝔠_{K} is anisotropic. We have

𝔡 = x𝔠_{K} ≃ x(u + v)𝔠_{K} = c𝔠_{K}.

Therefore, 0 = s_{∗}(𝔡) = 𝔟 in W (F) as needed. □

Corollary 35.13. Let K/F be a finite normal extension with F not formally real. If I^{n}(K) = 0 for some n, then I^{n}(F) = 0.

The above corollary is, in fact, true without the condition that K/F be normal. We have already shown this to be the case for fields of characteristic 2 in Lemma 35.5 and will show it to be true for characteristic not 2 in Corollary 43.9 below. In fact, it follows from §43 below that if K/F a finitely generated field extension of transcendence degree m, then I^{n}(K) = 0 implies n ≥ m and I^{n−m}(F) = 0. But we cannot do this by the elementary means employed here.

Corollary 35.14. Let K/F be a quadratic extension.

(1) Suppose that I^{n}(K) = 0. Then L satisfies A_{n} for every extension L/F such that [L : F] ≤ 2.

(2) Suppose that I^{n}(K) = 0. Then I^{n}(F) = ⟨⟨−w⟩⟩I^{n−1}(F) for every w ∈ D(∞⟨1⟩).

(3) Suppose that I^{n}(F) = ⟨⟨−w⟩⟩I^{n−1}(F) for some w ∈ F^{×}. Then both F and K satisfy A_{n+1} and if char F ≠ 2, then w ∈ D(∞⟨1⟩).

Proof. (1), (2): By Corollaries 35.8 and 35.13 if F is not formally real, then I^{n}(F) = 0 if and only if I^{n}(L) = 0 for any quadratic extension L/F. In particular,

(1) and (2) follow if F is not formally real. So suppose that F is formally real. We may assume that K = F (\sqrt{a}) with a ∈ F^{×}. Then I^{n}(L(\sqrt{a})) = 0 by Proposition 35.7, hence I^{n}(L) satisfies A_{n} by Theorem 35.12. This establishes (1).

Let w ∈ D(∞⟨1⟩). Then F (\sqrt{−w}) is not formally real. By (1), the field F (\sqrt{−w}) satisfies A_{n}, hence I^{n}(F (\sqrt{−w})) = 0. In particular, if 𝔟 is a bilinear n-fold Pfister form, then 𝔟_{F (\sqrt{−w})} is metabolic. Thus 𝔟 ≃ ⟨⟨−w⟩⟩ ⊗ 𝔠 for some (n − 1)-fold Pfister form 𝔠 over F by Remark 34.23 and (2) follows.

(3): If char F = 2, then I^{n}(F) = 0, hence I^{n}(K) = 0 by Corollary 35.8. So we may assume that char F ≠ 2. By Remark 34.23, we have 2^{n}⟨1⟩ ≃ ⟨⟨−w⟩⟩ ⊗ 𝔟 for some bilinear (n − 1)-fold Pfister form 𝔟. As 2^{n}⟨1⟩ only represents elements in D̃(∞⟨1⟩), we have w ∈ D(∞⟨1⟩).

To show the first statement, it suffices to show that L = F (\sqrt{−w}) satisfies A_{n+1} by (1) and (2). Since I^{n+1}(L) is generated by Pfister forms of the type ⟨⟨x⟩⟩ ⊗ 𝔠_{L} where x ∈ L^{×} and 𝔠 is an n-fold Pfister form over F by Lemma 34.16, we have I^{n+1}(L) ⊂ ⟨⟨−w⟩⟩I^{n}(L) = {0}. □

If F is the field of 2-adic numbers, then I^{2}(F) = 2I(F) and K satisfies I^{3}(K) = 0 for all finite extensions K/F but no such K satisfies I^{2}(K) = 0. In particular, statement (3) of Corollary 35.14 is the best possible.

Corollary 35.15. Let F be a field extension of transcendence degree n over a real closed field. Then D(2^{n}⟨1⟩) = D(∞⟨1⟩).

Proof. As F (\sqrt{−1}) is a C_{n}-field by Theorem 97.7 below, we have

I^{n}(F (\sqrt{−1})) = 0.

Therefore, F satisfies A_{n} by Corollary 35.14. □

35.B. Torsion-freeness and I^{n}(F (\sqrt{−1})) = 0. We intend to prove a result of Krüskemper 35.26 (cf. [87]) showing F satisfying A_{n} and I^{n}(F) torsion-free are equivalent when I^{n}(F (\sqrt{−1})) = 0. We follow Arason’s notes (cf. [3]) generalizing our congruence relations of Pfister forms developed in §24.

Let 𝔟 be a bilinear Pfister form. For simplicity, we set

I_{𝔟}(F) = {𝔠 ∈ I(F) | 𝔟 · 𝔠 = 0 ∈ W (F)} = I(F) ∩ ann_{W (F)}(𝔟) ⊂ I(F).

We note if 𝔟 is metabolic, then I_{𝔟}(F) = I(F). We tacitly assume that 𝔟 is anisotropic below.

Lemma 35.16. Let 𝔠 be a bilinear (n − 1)-fold Pfister form and d ∈ D_{F} (𝔟 ⊗ 𝔠). Then ⟨⟨d⟩⟩ · 𝔠 ∈ I^{n−1}(F)I_{𝔟}(F).

Proof. We induct on n. The hypothesis implies that 𝔟 · ⟨⟨d⟩⟩ · 𝔠 = 0 in W (F), hence ⟨1, −d⟩ · 𝔠 ∈ I_{𝔟}(F). In particular, the case n = 1 is trivial. So assume that n > 1 and that the lemma holds for (n − 2)-fold Pfister forms. Write 𝔠 = ⟨⟨a⟩⟩ ⊗ 𝔡 where 𝔡 is an (n − 2)-fold Pfister form. Then d = e_{1} − ae_{2}, where e_{1}, e_{2} ∈ D̃(𝔟 ⊗ 𝔡). If e_{2} = 0, then we are done by the induction hypothesis. So assume that e_{2} ≠ 0. Then d = e_{2}(e − a), where e = e_{1}/e_{2} ∈ D̃(𝔟 ⊗ 𝔡). By the induction hypothesis, we have

⟨⟨d⟩⟩ · 𝔠 = ⟨⟨e_{2}(e − a)⟩⟩ · 𝔠 = ⟨⟨e − a⟩⟩ · 𝔠 + ⟨⟨e − a, e_{2}⟩⟩ · 𝔠
≡ ⟨⟨e − a⟩⟩ · 𝔠 mod I^{n−1}(F)I_{𝔟}(F).

It follows that we may assume that e_{2} = 1, hence that d = e − a. But then

⟨⟨d, a⟩⟩ = ⟨⟨e − a, a⟩⟩ = ⟨⟨e, a′⟩⟩

for some a′ ≠ 0 by Lemma 4.15 , hence

⟨⟨d⟩⟩ · 𝔠 = ⟨⟨d, a⟩⟩ · 𝔡 = ⟨⟨e, a′⟩⟩ · 𝔡.

By the induction hypothesis, it follows that ⟨⟨d⟩⟩ · 𝔠 ∈ I^{n−1}(F)I_{𝔟}(F). □

Lemma 35.17. Let 𝔢 be a bilinear n-fold Pfister form and b ∈ D(𝔟⊗𝔢′). Then there is a bilinear (n − 1)-fold Pfister form 𝔣 such that 𝔢 ≡ ⟨⟨b⟩⟩ · 𝔣 mod I^{n−1}(F)I_{𝔟}(F).

Proof. We induct on n. If n = 1, then 𝔢′ = ⟨⟨a⟩⟩ and b = ax for some x ∈ D(−𝔟). It follows that

⟨⟨b⟩⟩ = ⟨⟨ax⟩⟩ = ⟨⟨a⟩⟩ + a⟨⟨x⟩⟩ ≡ ⟨⟨a⟩⟩ mod I_{𝔟}(F).

Now assume that n > 1 and that the lemma holds for (n−1)-fold Pfister forms. Write 𝔢 = ⟨⟨a⟩⟩ ⊗ 𝔡 with 𝔡 an (n − 1)-fold Pfister form. Then b = c + ad, where c ∈ D̃(𝔟 ⊗ 𝔡′) and d ∈ D̃(𝔟 ⊗ 𝔡). If d = 0, then we are done by the induction hypothesis. So assume that d ≠ 0. Then

⟨⟨ad⟩⟩ · 𝔡 = ⟨⟨a⟩⟩ · 𝔡 + a⟨⟨d⟩⟩ · 𝔡
≡ ⟨⟨a⟩⟩ · 𝔡 mod I^{n−1}(F)I_{𝔟}(F)

by Lemma 35.16. It follows that we may assume that d = 1, hence b = c + a. If c = 0, then b = a and there is nothing to prove. So assume that c ≠ 0. By the induction hypothesis, we can write

𝔡 ≡ ⟨⟨c⟩⟩ · 𝔤 mod I^{n−2}(F)I_{𝔟}(F)

with 𝔤 an (n − 2)-fold Pfister form. As

⟨⟨a, c⟩⟩ = ⟨⟨b − c, c⟩⟩ ≃ ⟨⟨b, c′⟩⟩

for some c′ ≠ 0 by Lemma 4.15, it follows that

𝔢 = ⟨⟨a⟩⟩ · 𝔡 ≡ ⟨⟨a, c⟩⟩ ⊗ 𝔤
= ⟨⟨b, c′⟩⟩ · 𝔤 mod I^{n−1}(F)I_{𝔟}(F)

as needed. □

Lemma 35.18. Let 𝔢 be a bilinear n-fold Pfister form and 𝔥 a bilinear form over F.

(1) If 𝔢 ∈ I_{𝔟}(F), then 𝔢 ∈ I^{n−1}(F)I_{𝔟}(F).

(2) If 𝔥 · 𝔢 ∈ I_{𝔟}(F), then 𝔥 · 𝔢 ∈ I^{n−1}(F)I_{𝔟}(F).

Proof. (1): The hypothesis implies that 𝔟 · 𝔢 = 0 in W (F). In particular, 𝔟 ⊗ 𝔢 = 𝔟 ⊥ 𝔟 ⊗ 𝔢′ is isotropic. It follows that there exists an element b ∈ D_{F} (𝔟) ∩ D_{F} (𝔟 ⊗ 𝔢′). By Lemma 35.17,

𝔢 ≡ ⟨⟨b⟩⟩ · 𝔣 ≡ 0 mod I^{n−1}(F)I_{𝔟}(F).

(2): The hypothesis implies that 𝔥 · 𝔟 · 𝔢 = 0 in W (F). If 𝔟 · 𝔢 = 0 in W (F) then, by (1), we have 𝔢 ∈ I^{n−1}(F)I_{𝔟}(F) and we are done. Otherwise, we have 𝔥 ∈ I_{𝔟⊗𝔢}(F), which is generated by the forms ⟨⟨x⟩⟩, with x ∈ D(𝔟 ⊗ 𝔢). It therefore suffices to prove the claim in the case 𝔥 = ⟨⟨x⟩⟩. But then, by (1), we even have 𝔥 · 𝔢 ∈ I^{n}(F)I_{𝔟}(F). □

Lemma 35.19. Suppose the bilinear n-fold Pfister forms 𝔢, 𝔣 satisfy

a𝔢 ≡ b𝔣 mod I_{𝔟}(F)

with a, b ∈ F^{×}. Then

a𝔢 ≡ b𝔣 mod I^{n−1}(F)I_{𝔟}(F).

Proof. We induct on n. As the case n = 1 is trivial, we may assume that n > 1 and that the claim holds for (n−1)-fold Pfister forms. The hypothesis implies that a𝔟 ⊗ 𝔢 ≃ b𝔟 ⊗ 𝔣, in particular, b/a ∈ D_{F} (𝔟 ⊗ 𝔢). By Lemma 35.16, we therefore have

a𝔢 ≡ b𝔣 mod I^{n−1}(F)I_{𝔟}(F)

(actually, mod I^{n}(F)I_{𝔟}(F)). Hence we may assume that a = b. Dividing by a, we may even assume that a = b = 1. Write

𝔢 = ⟨⟨c⟩⟩ ⊗ 𝔤 and 𝔣 = ⟨⟨d⟩⟩ ⊗ 𝔥

with 𝔤, 𝔥 being (n − 1)-fold Pfister forms. The hypothesis now implies that 𝔟 ⊗ 𝔢′ ≃ 𝔟 ⊗ 𝔣′. In particular, d ∈ D(𝔟 ⊗ 𝔢′). By Lemma 35.17, we can write 𝔢 ≡ ⟨⟨d⟩⟩ · 𝔤_{1} mod I^{n−1}(F)I_{𝔟}(F) with 𝔤_{1} an (n − 1)-fold Pfister form. It follows that we may assume that c = d. By the induction hypothesis, we then have 𝔤 ≡ 𝔥 mod I^{n−2}(F)I_{𝔟⊗⟨⟨d⟩⟩}(F), hence

⟨⟨d⟩⟩ · 𝔤 ≡ ⟨⟨d⟩⟩ · 𝔥 mod ⟨⟨d⟩⟩I^{n−2}(F)I_{𝔟⊗⟨⟨d⟩⟩}(F).

We are therefore finished if we can show that ⟨⟨d⟩⟩I_{𝔟⊗⟨⟨d⟩⟩}(F) ⊆ I(F)I_{𝔟}(F). Now, I_{𝔟⊗⟨⟨d⟩⟩}(F) is generated by the ⟨⟨x⟩⟩, with x ∈ D(𝔟 ⊗ ⟨⟨d⟩⟩). For such a generator ⟨⟨x⟩⟩, we have 𝔟 · ⟨⟨d, x⟩⟩ = 0 in W (F), hence, by Lemma 35.18, the form ⟨⟨d, x⟩⟩ lies in I(F)I_{𝔟}(F). □

Proposition 35.20. Let 𝔢, 𝔣, 𝔤 be bilinear n-fold Pfister forms. Assume that

a𝔢 ≡ b𝔣 + c𝔤 mod I_{𝔟}(F).

Then

a𝔢 ≡ b𝔣 + c𝔤 mod I^{n−1}(F)I_{𝔟}(F).

Proof. The hypothesis implies that a𝔟·𝔢 = b𝔟·𝔣+c𝔟·𝔤 in W (F). In particular, the form b𝔟 ⊗ 𝔣 ⊥ c𝔟 ⊗ 𝔤 is isotropic. It follows that there exists d ∈ D(b𝔟 ⊗ 𝔣) ∩ D(−c𝔟 ⊗ 𝔤). By Lemma 35.16, we then have

b𝔣 ≡ d𝔣 mod I^{n−1}(F)I_{𝔟}(F) and c𝔤 ≡ −d𝔤 mod I^{n−1}(F)I_{𝔟}(F)

(actually, mod I^{n}(F)I_{𝔟}(F)). Hence we may assume that c = −b. Dividing by b, we may even assume that b = 1 and c = −1. Then the hypothesis implies that a𝔟·𝔢 = 𝔟·𝔣−𝔟·𝔤 in W (F) and we have to prove that a𝔢 ≡ 𝔣−𝔤 mod I^{n−1}(F)I_{𝔟}(F).

As a𝔟 · 𝔢 = 𝔟 · 𝔣 − 𝔟 · 𝔤 in W (F), it follows that 𝔟 ⊗ 𝔣 and 𝔟 ⊗ 𝔤 are linked using Proposition 6.21 and with 𝔟 dividing the linkage. Hence there exists an (n − 1)- fold Pfister form 𝔡 and elements b′, c′ ≠ 0 such that 𝔟 ⊗ 𝔣 ≃ 𝔟 ⊗ 𝔡 ⊗ ⟨⟨b′⟩⟩ and 𝔟 ⊗ 𝔤 ≃ 𝔟 ⊗ 𝔡 ⊗ ⟨⟨c′⟩⟩ (and hence 𝔟 ⊗ 𝔢 ≃ 𝔟 ⊗ 𝔡 ⊗ ⟨⟨b′c′⟩⟩). By Lemma 35.19, we then have

𝔣 ≡ 𝔡 · ⟨⟨b′⟩⟩ and also 𝔤 ≡ 𝔡 · ⟨⟨c′⟩⟩ mod I^{n−1}(F)I_{𝔟}(F).

We may therefore assume that 𝔣 = 𝔡 ⊗ ⟨⟨b′⟩⟩ and 𝔤 = 𝔡 ⊗ ⟨⟨c′⟩⟩. Then 𝔣 − 𝔤 = 𝔡·⟨−b′, c′⟩ = −b′𝔡·⟨⟨b′c′⟩⟩ in W (F). The result now follows from Lemma 35.19. □

Remark 35.21. From Lemmas 35.16-35.19 and Proposition 35.20, we easily see that the corresponding results hold for the torsion part I_{t}(F) of I(F) instead of I_{𝔟}(F). Indeed, in each case, we only have to use our result for 𝔟 = 2^{k}⟨1⟩ for some k ≥ 0.

We always have 2I^{n}(F) ⊂ I^{n+1}(F) for a field F. For some interesting fields, we have equality, i.e., 2I^{n}(F) = I^{n+1}(F) for some positive integer n. In particular, we shall see in Lemma 41.1 below that this is true for some n for any field of finite transcendence degree over its prime field. (This is easy if the field has positive characteristic but depends on Fact 16.2 when the characteristic of F is 0.) We shall now investigate when this phenomenon holds for a field.

Proposition 35.22. Let F be a field. Then 2I^{n}(F) = I^{n+1}(F) if and only if every anisotropic bilinear (n + 1)-fold Pfister form 𝔟 is divisible by 2⟨1⟩, i.e., 𝔟 ≃ 2𝔠 for some n-fold Pfister form 𝔠.

Proof. If 2⟨1⟩ is metabolic, the result is trivial so assume not. In particular, we may assume that char F ≠ 2. Suppose 2I^{n}(F) = I^{n+1}(F) and 𝔟 is an anisotropic bilinear (n + 1)-fold Pfister form. By assumption, there exist 𝔡 ∈ I^{n}(F) such that 𝔟 = 2𝔡 in W (F). By Remark 34.23, we have 𝔟 ≃ 2𝔠 for some n-fold Pfister form 𝔠. □

It is also useful to study a variant of the property that 2I^{n}(F) = I^{n+1}(F) introduced by Bröcker in [19]. Recall that I_{red}^{n}(F) is the image of I^{n}(F) under the canonical homomorphism W (F) → W_{red}(F) = W (F)/W_{t}(F). We investigate the case that 2I_{red}^{n}(F) = I_{red}^{n+1}(F) for some positive integer n. Of course, if 2I^{n}(F) = I^{n+1}(F), then 2I_{red}^{n}(F) = I_{red}^{n+1}(F). We shall show that the above proposition generalizes. Further, we shall show this property is characterized by the cokernel of the signature map

sgn : W (F) → C(𝔛(F), ℤ).

Recall that the cokernel of this map is a 2-primary group by Theorem 33.8.

Proposition 35.23. Suppose the reduced stability st_{r}(F) is finite and equals n. Then n is the least nonnegative integer such that 2I_{red}^{n}(F) = I_{red}^{n+1}(F). Moreover, for any bilinear (n + 1)-fold Pfister form 𝔟, there exists an n-fold Pfister form 𝔠 such that 𝔟 ≡ 2𝔠 mod W_{t}(F).

Proof. Let 𝔟 be an anisotropic bilinear (n+1)-fold Pfister form. In particular, sgn 𝔟 ∈ C(𝔛(F), 2^{n+1}ℤ). By assumption, there exists a bilinear form 𝔡 satisfying sgn 𝔡 = \frac{1}{2} sgn 𝔟. Thus 𝔟 − 2𝔡 ∈ W_{t}(F), hence there exists an integer m such that 2^{m}𝔟 = 2^{m+1}𝔡 in W (F) by Theorem 31.18. If 2^{m}𝔟 is metabolic the result is trivial, so we may assume it is anisotropic. By Proposition 6.22, there exists 𝔣 such that 2^{m}𝔟 ≃ 2^{m+1}𝔣. Therefore, 2^{m}𝔟 ≃ 2^{m+1}𝔠 for some bilinear n-fold Pfister form 𝔠 by Corollary 6.17. Hence 2I_{red}^{n}(F) = I_{red}^{n+1}(F).

Conversely, suppose that 2I_{red}^{n}(F) = I_{red}^{n+1}(F). Let f ∈ C(𝔛(F), ℤ). It suffices to show that there exists a bilinear form 𝔟 satisfying sgn 𝔟 = 2^{n}f. By Theorem 33.15, there exists an integer m and a bilinear form 𝔟 ∈ I^{m}(F) satisfying sgn 𝔟 = 2^{m}f. So we are done if m ≤ n. If m > n, then there exists 𝔠 ∈ I^{n}(F) such that sgn 𝔟 = sgn 2^{m−n}𝔠 and 2^{n}f = sgn 𝔠. □

Remark 35.24. If 2I_{red}^{n}(F) = I_{red}^{n+1}(F), then for any bilinear (n + m)-fold Pfister form 𝔟 there exists an n-fold Pfister form 𝔠 such that 𝔟 ≡ 2^{m}𝔠 mod I_{t}(F) and I_{red}^{n+m}(F) = 2^{m}I_{red}^{n}(F). Similarly, if 2I^{n}(F) = I^{n+1}(F), then for any bilinear (n + m)-fold Pfister form 𝔟 there exists an n-fold Pfister form 𝔠 such that 𝔟 ≃ 2^{m}𝔠 and I^{n+m}(F) = 2^{m}I^{n}(F).

Suppose that 2I_{red}^{n}(F) = I_{red}^{n+1}(F). Let 𝔟 be an n-fold Pfister form over F and let d ∈ F^{×}. Write

⟨⟨d⟩⟩ · 𝔟 ≡ 2𝔢 mod I_{t}(F) and ⟨⟨−d⟩⟩ · 𝔟 ≡ 2𝔣 mod I_{t}(F)

for some n-fold Pfister forms 𝔢 and 𝔣 over F. By adding, we then get 2𝔟 ≡ 2𝔢 + 2𝔣 mod I_{t}(F), hence also 𝔟 ≡ 𝔢 + 𝔣 mod I_{t}(F). By Proposition 35.20, it follows that we even have 𝔟 ≡ 𝔢 + 𝔣 mod I^{n−1}(F)I_{t}(F).

We generalize this as follows:

Lemma 35.25. Suppose that 2I_{red}^{n}(F) = I_{red}^{n+1}(F). Let 𝔟 be a bilinear n-fold Pfister form and let d_{1}, …, d_{m} ∈ F^{×}. Write

⟨⟨ε_{1}d_{1}, …, ε_{m}d_{m}⟩⟩ · 𝔟 ≡ 2^{m}𝔠_{ε} mod I_{t}(F)

with 𝔠_{ε} a bilinear n-fold Pfister form for every ε = (ε_{1}, …, ε_{m}) ∈ {±1}^{m}. Then

𝔟 ≡ ∑_{ε} 𝔠_{ε} mod I^{n−1}(F)I_{t}(F).

Proof. We induct on m. The case m = 1 is done above. So assume that m > 1. Write ⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩·𝔟 ≡ 2^{m−1}𝔡_{ε′} mod I_{t}(F) with 𝔡_{ε′} a bilinear n-fold Pfister form for every ε′ = (ε_{2}, …, ε_{m}) ∈ {±1}^{m−1}. By the induction hypothesis, we then have 𝔟 ≡ ∑_{ε′} 𝔡_{ε′} mod I^{n−1}(F)I_{t}(F). It therefore suffices to show that

𝔡_{ε′} ≡ 𝔠_{(+1,ε′)} + 𝔠_{(−1,ε′)} mod I^{n−1}(F)I_{t}(F)

for every ε′. Since

2^{m}𝔡_{ε′} ≡ 2⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩ · 𝔠 = (⟨⟨d⟩⟩ + ⟨⟨−d⟩⟩) · ⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩ · 𝔢
≡ 2^{m}𝔠_{(+1,ε′)} + 2^{m}𝔠_{(−1,ε′)} mod I_{t}(F)

in W (F), hence also 𝔡_{ε′} ≡ 𝔠_{(+1,ε′)} + 𝔠_{(−1,ε′)} mod I_{t}(F). By Proposition 35.20, it follows that 𝔡_{ε′} ≡ 𝔠_{(+1,ε′)} + 𝔠_{(−1,ε′)} mod I^{n−1}(F)I_{t}(F). □

Krüskemper’s main result in [87] is:

Theorem 35.26. Let 2I_{red}^{n}(F) = I_{red}^{n+1}(F). Then

I_{t}^{n}(F) = I^{n−1}(F)I_{t}(F).

Proof. Suppose that ∑_{i=1}^{r} a_{i}𝔟_{i} ∈ I_{t}(F), where 𝔟_{1}, …, 𝔟_{r} are bilinear n-fold Pfister forms and a_{i} ∈ F^{×}. We prove by induction on r that this implies that ∑_{i=1}^{r} a_{i}𝔟_{i} ∈ I^{n−1}(F)I_{t}(F). The case r = 1 is simply Lemma 35.18, so assume that r > 1.

Write 𝔟_{i} = ⟨⟨a_{i1}, …, a_{in}⟩⟩ for i = 1, …, r and let m = rn and
(d_{1}, …, d_{m}) = (a_{11}, …, a_{1n}, a_{21}, …, a_{2n}, …, a_{r1}, …, a_{rn}).

Write

⟨⟨ε_{1}d_{1}, …, ε_{m}d_{m}⟩⟩ · 𝔟_{i} ≡ 2^{m}𝔠_{iε} mod I_{t}(F)

with 𝔠_{iε} bilinear n-fold Pfister forms for every i = 1, …, r and every

ε = (ε_{1}, …, ε_{m}) ∈ {±1}^{m}.

By Lemma 35.25,

∑_{i=1}^{r} a_{i}𝔟_{i} ≡ ∑_{ε} ∑_{i=1}^{r} a_{i}𝔠_{iε} mod I^{n−1}(F)I_{t}(F).

If ε^{(1)} ≠ ε^{(2)} in {±1}^{m}, then sgn⟨⟨ε_{1}^{(1)}d_{1}, …, ε_{m}^{(1)}d_{m}⟩⟩ and sgn⟨⟨ε_{1}^{(2)}d_{1}, …, ε_{m}^{(2)}d_{m}⟩⟩ have disjoint supports on 𝔛(F), hence the same holds for sgn 𝔠_{iε^{(1)}} and sgn 𝔠_{jε^{(2)}}. It therefore follows from the hypothesis that

∑_{i=1}^{r} a_{i}𝔠_{iε} ≡ 0 mod I_{t}(F) for each ε.

Clearly, it suffices to show that ∑_{i=1}^{r} a_{i}𝔠_{iε} ≡ 0 mod I^{n−1}(F)I_{t}(F) for each ε.

Fix ε in {±1}^{m}. Suppose that ε ≠ (1, …, 1). If −1 occurs as the coordinate of ε corresponding to a_{ji} for some j ∈ [1, r] and i ∈ [1, n], then ⟨⟨ε_{1}d_{1}, …, ε_{m}d_{m}⟩⟩·𝔟_{j} = 0 in W (F) and we may assume for all such j that 𝔠_{jε} = 0 in W (F). In particular, if ε ≠ (1, …, 1), then

∑_{i=1}^{r} a_{i}𝔠_{iε} = ∑_{\begin{matrix}i=1 \\ i≠j \end{matrix}}^{r} a_{i}𝔠_{iε} ≡ 0 mod I^{n−1}(F)I_{t}(F)

by the induction hypothesis. So we may assume that ε = (1, …, 1). Then

⟨⟨ε_{1}d_{1}, …, ε_{m}d_{m}⟩⟩ ⊗ 𝔟_{i} ≃ ⟨⟨d_{1}, …, d_{m}⟩⟩ ⊗ 𝔟_{i} ≃ 2^{n}⟨⟨d_{1}, …, d_{m}⟩⟩

is independent of i. We therefore may assume that 𝔠_{iε}, for i ∈ [1, r], are all equal to a single 𝔠. Let 𝔡 = ⟨a_{1}, …, a_{r}⟩, then

𝔡 · 𝔠 = ∑_{i=1}^{r} a_{i}𝔠_{iε} ≡ 0 mod I_{t}(F).

By Lemma 35.18, we conclude that 𝔡·𝔠 ∈ I^{n−1}(F)I_{t}(F) and the theorem follows. □

Corollary 35.27. The following are equivalent for a field F of characteristic different from 2:

(1) I^{n+1}(F (\sqrt{−1})) = 0.

(2) F satisfies A_{n+1} and 2I^{n}(F) = I^{n+1}(F).

(3) F satisfies A_{n+1} and 2I_{red}^{n}(F) = I_{red}^{n+1}(F).

(4) I^{n+1}(F) is torsion-free and 2I^{n}(F) = I^{n+1}(F).

Proof. (1) ⇒ (2): By Theorem 35.12, F satisfies A_{n+1}. Theorem 34.22 applied to the quadratic extension F (\sqrt{−1})/F gives 2I^{n}(F) = I^{n+1}(F).

(2) ⇒ (3) is trivial as 2I_{red}^{n}(F) = I_{red}^{n+1}(F) if 2I^{n}(F) = I^{n+1}(F).

(3) ⇒ (4): As the torsion (n + 1)-fold Pfister forms generate the torsion in I^{n+1}(F) by Theorem 35.26, we have I^{n+1}(F) is torsion-free. Suppose that 𝔟 is an (n + 1)-fold Pfister form. Then there exist 𝔠 ∈ I^{n}(F) and 𝔡 ∈ W_{t}(F) such that 𝔟 = 2𝔠 + 𝔡 in W (F). Hence for some N, we have 2^{N}𝔟 = 2^{N+1}𝔠. As I^{n+1}(F) is torsion-free, we have 𝔟 = 2𝔠 in W (F), hence 𝔟_{F (\sqrt{−1})} is hyperbolic. By Theorem 34.22, there exists an n-fold Pfister form 𝔣 such that 𝔟 ≃ 2𝔣. It follows that 2I^{n}(F) = I^{n+1}(F).

(4) ⇒ (1) follows from Theorem 34.22 for the quadratic extension F (\sqrt{−1})/F as forms in W (K) transfer to torsion forms in W (F). □

Remark 35.28. By Corollary 35.14, the condition that I^{n+1}(F (\sqrt{−1})) = 0 is equivalent to I^{n+1}(K) = 0 for some quadratic extension K/F. In particular, if I^{n+1}(K) = 0 for some quadratic extension K/F, then I^{n+1}(F) is torsion-free. Much more is true. If K/F is a field extension of transcendence degree m, then I^{n}(K) torsion-free implies n ≥ m and I^{n−m}(F) is torsion-free. We shall prove this in Corollary 43.9 below.

Corollary 35.29. Let F be a real closed field and K/F a finitely generated extension of transcendence degree n. Then I^{n+1}(K) is torsion-free and 2I^{n}(K) = I^{n+1}(K).

Proof. As K(\sqrt{−1}) is a C_{n}-field by Theorem 97.7, we have I^{n+1}(K(\sqrt{−1})) = 0 and hence 2I^{n}(K) = I^{n+1}(K) by Corollary 35.27 applied to the field K. □

Corollary 35.30. Let F be a field satisfying I^{n+1}(F) = 2I^{n}(F). Then I^{n+2}(F) is torsion-free.

Proof. If −1 ∈ F^{2}, then I^{n+1}(F) = 0 and the result follows. In particular, we may assume that char F ≠ 2. By Theorem 35.26, it suffices to show that F satisfies A_{n+2}. Let 𝔟 be an (n + 2)-fold Pfister form such that 2𝔟 = 0 in W (F). By Lemma 35.2, we can write 𝔟 = ⟨⟨w⟩⟩ · 𝔠 in W (F) with 𝔠 an (n + 1)-fold Pfister form and w ∈ D(2⟨1⟩). By assumption, 𝔠 = 2𝔡 in W (F) for some n-fold Pfister form 𝔡. Hence 𝔟 = 2⟨⟨w⟩⟩ · 𝔡 = 0 in W (F). □

Remark 35.31. Any local field F satisfies I^{3}(F) = 0 (cf. [89, Cor. VI.2.15]). Let ℚ_{2} be the field of 2-adic numbers. Then, up to isomorphism, \binom{−1,−1}{ℚ_{2}} is the unique

quaternion algebra (cf. [89, Cor. VI.2.24]), hence I^{2}(ℚ_{2}) = 2I(ℚ_{2}) = {0, 4⟨1⟩} ≠ 0. Thus, in general, I^{n+2}(F) cannot be replaced by I^{n+1}(F) in the corollary above.

We shall return to these matters in §41.

CHAPTER VI u-invariants

36. The ū-invariant

Given a field F, it is interesting to see if there exists a uniform bound on the dimension of anisotropic forms over F, i.e., if there exists an integer n such that every quadratic form over F of dimension greater than n is isotropic and if such an n exists, what is the minimum. For example, a consequence of the Chevalley-Warning Theorem (cf. [123, I.2, Th. 3]), is that over a finite field every 3-dimensional quadratic form is isotropic and a consequence of the Lang-Nagata Theorem (cf. Theorem 97.7 below) is that every (2^{n} + 1)-dimensional form over a field of transcendence degree n over an algebraically closed field is isotropic. When a field is of characteristic different from 2, anisotropic forms are nondegenerate. This is no longer the case for fields of characteristic 2. This leads to the consideration of two invariants. If F is a formally real field, then n⟨1⟩ can never be isotropic. To obtain meaningful arithmetic data about formally real fields, we shall strengthen the condition on our forms. Although this makes computation more delicate, it is a useful generalization. In this section, we shall, for the most part, look at the simpler case of fields that are not formally real.

Let F be a field. We call a quadratic form φ over F locally hyperbolic if φ_{F_{P}} is hyperbolic at each real closure F_{P} of F (if any). If F is formally real, then the dimension of every locally hyperbolic form is even. If F is not formally real, every form is locally hyperbolic. For any field F, a locally hyperbolic form is one that is torsion in the Witt ring W (F).

Define the u-invariant of F to be the smallest integer u(F) ≥ 0 such that every nondegenerate locally hyperbolic quadratic form over F of dimension > u(F) is isotropic (or infinity if no such integer exists) and the ū-invariant of F to be the smallest integer ū(F) ≥ 0 such that every locally hyperbolic quadratic form over F of dimension > ū(F) is isotropic (or infinity if no such integer exists). The u-invariant above was first defined in [34] and the ū-invariant by Baeza in [16].

Remark 36.1. (1) We have ū(F) ≥ u(F).

(2) If char F ≠ 2, every anisotropic form is nondegenerate, hence ū(F) = u(F).

(3) If F is formally real, the integer ū(F) = u(F) is even.

(4) As any (nondegenerate) quadratic form contains (nondegenerate) subforms of all smaller dimensions, if F is not formally real, we have u(F) ≤ n if and only if every nondegenerate quadratic form of dimension n + 1 is isotropic and ū(F) ≤ n if and only if every quadratic form of dimension n + 1 is isotropic.

Example 36.2. (1) If F is a formally real field, then ū(F) = 0 if and only if F is pythagorean.

162 VI. u-INVARIANTS

(2) Suppose that F is a quadratically closed field. If char F ≠ 2, then ū(F) = 1 as every form is diagonalizable. If char F = 2, then ū(F) ≤ 2 with equality if F is not separably closed by Example 7.33.

(3) If F is a finite field, then ū(F) = 2.

(4) Let F be a field of characteristic not 2 and ρ a quadratic form over the Laurent series field F ((t)). As ρ is diagonalizable, ρ ≅ φ ⊥ tψ for some quadratic forms φ and ψ over F. Using Lemma 19.5, we deduce that ρ is anisotropic if and only if both φ and ψ are. Moreover, one can check that ρ is torsion if and only if both φ and ψ are torsion. If ū(F) is finite, it follows that ū(F ((t))) = 2ū(F). This equality is also true if char F = 2 (cf. [16]).

(5) If F is a C_{n}-field, then ū(F) ≤ 2^{n}.

(6) If F is a local field, then ū(F) = 4. If char F > 0, then u(F) = 4 by Example (4). If char F = 0 this follows as in the argument for Example (4) using also Lemma 19.4 (cf. also [111]).

(7) If F is a global field, then ū(F) = 4. If char F = 0, this follows from the Hasse-Minkowski Theorem [89, VI.3.1]. If char F > 0, then F is a C_{2}-field by Theorem 97.7.

Proposition 36.3. Let F be a field with I_{q}^{3}(F) = 0. If 1 < u(F) < ∞, then u(F) is even.

Proof. The hypothesis implies that F is not formally real. Suppose that u(F) > 1 is odd and let φ be a nondegenerate anisotropic quadratic form with dim φ = u(F). We claim that φ ≃ ψ ⊥ ⟨−a⟩ for some ψ ∈ I_{q}^{2}(F) and a ∈ F^{×}. If char F ≠ 2, then φ ⊥ ⟨a⟩ ∈ I_{q}^{2}(F) for some a ∈ F^{×} by Proposition 4.13. This form is isotropic, hence φ ⊥ ⟨a⟩ ≃ ψ ⊥ ℍ for some ψ ∈ I_{q}^{2}(F) and therefore φ ≃ ψ ⊥ ⟨−a⟩. If char F = 2, write φ ≃ μ ⊥ ⟨a⟩ for some form μ and a ∈ F^{×}. Choose b ∈ F such that the discriminant of the form μ ⊥ [a, b] is trivial, i.e., μ ⊥ [a, b] ∈ I_{q}^{2}(F). By assumption the form μ ⊥ [a, b] is isotropic, i.e., μ ⊥ [a, b] ≃ ψ ⊥ ℍ for a form ψ ∈ I_{q}^{2}(F). It follows from (8.7) that

φ ≃ μ ⊥ ⟨a⟩ ∼ μ ⊥ [a, b] ⊥ ⟨a⟩ ∼ ψ ⊥ ⟨a⟩,

hence φ ≃ ψ ⊥ ⟨a⟩ as these forms have the same dimension. This proves the claim.

Let b ∈ D(ψ). As ⟨⟨ab⟩⟩ ⊗ ψ ∈ I_{q}^{3}(F) = 0, we have ab ∈ G(ψ). Therefore, a = ab/b ∈ D(ψ) and hence the form φ is isotropic, a contradiction. □

Corollary 36.4. The u-invariant of a field is not equal to 3, 5 or 7.

Let r > 0 be an integer. Define the ū_{r}-invariant of F to be the smallest integer ū_{r}(F) ≥ 0 such that every set of r quadratic forms on a vector space over F of dimension > ū_{r}(F) has common nontrivial zero.

In particular, if ū_{r}(F) is finite, then F is not a formally real field. We also have ū_{1}(F) = ū(F) when F is not formally real. The following is due to Leep (cf. [93]):

Theorem 36.5. Let F be a field. Then for every r > 1, we have

ū_{r}(F) ≤ rū_{1}(F) + ū_{r−1}(F).

Proof. We may assume that ū_{r−1}(F) is finite. Let φ_{1}, …, φ_{r} be quadratic forms on a vector space V over F of dimension n > rū_{1}(F) + ū_{r−1}(F). We shall show that the forms have an isotropic vector in V. Let W be a totally isotropic

36. THE ū-INVARIANT 163

subspace of V of the forms φ_{1}, …, φ_{r−1} of the largest dimension d. Let V_{i} be the orthogonal complement of W in V relative to φ_{i} for each i ∈ [1, r − 1]. We have dim V_{i} ≥ n − d.

Let U = V_{1} ∩ ⋯ ∩ V_{r−1}. Then W ⊂ U and dim U ≥ n − (r − 1)d. Choose a subspace U′ ⊂ U such that U = W ⊕ U′. We have

dim U′ ≥ n − rd > r(ū_{1}(F) − d) + ū_{r−1}(F).

If d ≤ ū_{1}(F), then dim U′ > ū_{r−1}(F), hence the forms φ_{1}, …, φ_{r−1} have an isotropic vector u ∈ U′. Then the subspace W ⊕ F u is totally isotropic for these forms, contradicting the maximality of W.

It follows that d > ū_{1}(F). The form φ_{r} therefore has an isotropic vector in U′ which is isotropic for all the φ_{i}’s. □

Corollary 36.6. If F is not formally real, then ū_{r}(F) ≤ \frac{1}{2}r(r + 1)ū(F).

This improves the linear bound in [31] to:

Corollary 36.7. Let K/F be a finite field extension of degree r. If F is not formally real, then ū(K) ≤ \frac{1}{2}(r + 1)ū(F).

Proof. Let s_{1}, s_{2}, …, s_{r} be a basis for the space of F -linear functionals on K. Let φ be a quadratic form over K of dimension n > \frac{1}{2}(r+1)ū(F). As dim(s_{i})_{∗}(φ) = rn > \frac{1}{2}r(r + 1)ū(F) for each i ∈ [1, r], by Corollary 36.6, the forms (s_{i})_{∗}(φ) have a common isotropic vector which is, then an isotropic vector for φ. □

Let K/F be a finite extension with F not formally real. We shall show that if ū(K) is finite, then so is ū(F). We begin with the case that F is a field of characteristic 2 with an observation by Mammone, Moresi, and Wadsworth (cf. [96]).

Lemma 36.8. Let F be a field of characteristic 2. Let φ be an even-dimensional nondegenerate quadratic form over F and ψ a totally singular quadratic form over F. If φ ⊥ ψ is anisotropic, then

\frac{1}{2} dim φ + dim ψ ≤ [F : F^{2}].

Proof. Let φ ≃ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{m}, b_{m}] with a_{i}, b_{i} ∈ F and ψ ≃ ⟨c_{1}, …, c_{n}⟩ with c_{i} ∈ F^{×}. For each i ∈ [1, m] let d_{i} ∈ D([a_{i}, b_{i}]). Then {c_{1}, …, c_{n}, d_{1}, …, d_{m}} is F^{2}-linearly independent. The result follows. □

Proposition 36.9. Let F be a field of characteristic 2 and K/F a finite extension. Then

ū(F) ≤ 2ū(K) ≤ 4ū(F).

Proof. If c_{1}, …, c_{n} are F^{2}-linearly independent, then the form ⟨c_{1}, …, c_{n}⟩ is anisotropic. By the lemma, it follows that we have

[F : F^{2}] ≤ ū(F) ≤ 2[F : F^{2}].

As [F : F^{2}] = [K : K^{2}] (cf. (35.6)), we have

ū(F) ≤ 2[F : F^{2}] = 2[K : K^{2}] ≤ 2ū(K) ≤ 4[K : K^{2}] = 4[F : F^{2}] ≤ 4ū(F). □

Remark 36.10. Let F be a field of characteristic 2. The proof above shows that every anisotropic totally singular quadratic form has dimension at most [F : F^{2}] and if [F : F^{2}] is finite, then there exists an anisotropic totally singular quadratic form of dimension [F : F^{2}].

Remark 36.11. Let F be a field of characteristic 2 such that [F : F^{2}] is infinite but F separably closed. Then ū(F) is infinite but u(F) = 1 by Exercise 7.34.

We now look at finiteness of ū coming down from a quadratic extension following the ideas in [34].

Proposition 36.12. Let K/F be a quadratic extension with F not formally real. If ū(K) is finite, then ū(F) < 4ū(K).

Proof. If char F = 2, then ū(F) ≤ 2ū(K) by Proposition 36.9, so we may assume that char F ≠ 2. We first show that ū(F) is finite. Let φ be an anisotropic quadratic form over F. By Proposition 34.8, there exist quadratic forms φ_{1} and μ_{0} over F with (μ_{0})_{K} anisotropic satisfying

φ ≃ ⟨⟨a⟩⟩ ⊗ φ_{1} ⊥ μ_{0}

where a ∈ F^{×} satisfies K = F (\sqrt{a}). In particular, dim(μ_{0}) ≤ ū(K). Analogously, there exist quadratic forms φ_{2} and μ_{1} over F with (μ_{1})_{K} anisotropic satisfying

φ_{1} ≃ ⟨⟨a⟩⟩ ⊗ φ_{2} ⊥ μ_{1}.

Hence

φ ≃ ⟨⟨a⟩⟩ ⊗ (⟨⟨a⟩⟩ ⊗ φ_{2} ⊥ μ_{1}) ⊥ μ_{0} ≃ 2⟨⟨a⟩⟩ ⊗ φ_{2} ⊥ ⟨⟨a⟩⟩ ⊗ μ_{1} ⊥ μ_{0}

as ⟨⟨a, a⟩⟩ = 2⟨⟨a⟩⟩. Continuing in this way, we see that

φ ≃ 2^{k−1}⟨⟨a⟩⟩ ⊗ φ_{k} ⊥ 2^{k−2}⟨⟨a⟩⟩ ⊗ μ_{k−1} ⊥ ⋯ ⊥ ⟨⟨a⟩⟩ ⊗ μ_{1} ⊥ μ_{0}

for some forms φ_{k} and μ_{i} over F satisfying dim μ_{i} ≤ ū(K) for all i. By Proposition 31.4, there exists an integer n such that 2^{n}⟨⟨a⟩⟩ = 0 in W (F). It follows that

dim φ ≤ (2^{n} + ⋯ + 2 + 1)ū(K) ≤ 2^{n+1}ū(K),

hence dim φ is finite.

We now show that ū(F) < 4ū(K). As ū(F) is finite, there exists an anisotropic form φ over F of dimension ū(F). Let s : K → F be a nontrivial F -linear functional satisfying s(1) = 0. We can write

φ ≃ μ ⊥ s_{∗}(ψ)

with quadratic forms ψ over K and μ over F satisfying μ ⊗ N_{K/F} is anisotropic by Proposition 34.6. Then

dim s_{∗}(ψ) ≤ 2ū(K) and dim μ ≤ ū(F)/2.

If dim s_{∗}(ψ) = 2ū(K), then ψ is a ū(K)-dimensional form over K, hence universal as every (ū(K)+1)-dimensional form is isotropic over the nonformally real field K. In particular, ψ ≃ ⟨x⟩_{K} ⊥ ψ_{1} for some x ∈ F^{×}. Thus s_{∗}(ψ) = s_{∗}(ψ_{1}) in W (F), so s_{∗}(ψ) is isotropic, a contradiction. Therefore, we have dim s_{∗}(ψ) < 2ū(K), hence

2ū(K) > dim s_{∗}(ψ) = dim φ − dim μ ≥ ū(F) − ū(F)/2 ≥ ū(F)/2.

The result follows. □

37. THE u-INVARIANT FOR FORMALLY REAL FIELDS 165

Proposition 36.13. Let K/F be a finite extension with F not formally real. Then ū(F) is finite if and only if ū(K) is finite.

Proof. If char F = 2, the result follows by Proposition 36.9, so we may assume that char F ≠ 2. By Corollary 36.7, we need only show if ū(K) is finite, then ū(F) is also finite. Let L be the normal closure of K/F and E_{0} the fixed field of the Galois group of L/F. Then E_{0}/F is of odd degree as char F ≠ 2. Let E be the fixed field of a Sylow 2-subgroup of the Galois group of L/F. Then E/F is also of odd degree. Therefore, if ū(E) is finite so is ū(F) by Springer’s Theorem 18.5. Hence we may assume that E = F. By Corollary 36.7, we have ū(L) is finite, so we may assume that L = K and K/F is a Galois 2-extension. By induction on [K : F], we may assume that K/F is a quadratic extension, the case established in Proposition 36.12. □

Let K/F be a normal extension of degree 2^{m}r with r odd and F not formally real. If ū(K) is finite, the argument in Proposition 36.13 and the bound in Proposition 36.12 shows that ū(F) ≤ 4^{r}ū(K). We shall improve this bound in Remark 37.6 below.

37. The u-invariant for formally real fields

If F is formally real and K/F finite, then ū(K) can be infinite and ū(F) finite. Indeed, let F_{0} be the euclidean field of real constructible numbers. Then there exist extensions E_{r}/F_{0} of degree r none of which are both pythagorean and formally real. In particular, ū(E_{r}) > 0. It is easy to see that ū(E_{r}) ≤ 4. (In fact, it can be shown that ū(E_{r}) ≤ 2.) For example, E_{2} is the quadratic closure of the rational numbers. Let F = F_{0}((t_{1})) ⋯ ((t_{n})) ⋯ denote the field of iterated Laurent series in infinitely many variables over F_{0}. Then F is pythagorean by Example 36.2(1) so ū(F) = 0. However, the field E_{r}((t_{1})) ⋯ ((t_{n})) ⋯ has infinite u-invariant by Example 36.2(4). In fact, in [42] for each positive integer n, formally real fields F_{n} are constructed with ū(F_{n}) = 2^{n} and having a formally real quadratic extension K/F_{n} with ū(K) = ∞ and formally fields F′_{n} are constructed with ū(F′_{n}) = 2^{n} and such that every finite nonformally real extension L of F has infinite ū-invariant.

However, we can determine when finiteness of the ū-invariant persists when going up a quadratic extension and when coming down one. Since we already know this when the base field is not formally real, we shall mostly be interested in the formally real case. In particular, we shall assume that the fields in this section are of characteristic different from 2 and hence the ū-invariant and u-invariant are identical.

We need some preliminaries.

Lemma 37.1. Let F be a field of characteristic different from 2 and K = F (\sqrt{a}) a quadratic extension of F. Let b ∈ F^{×} \ F^{×2} and φ ∈ ann_{W (F)}(⟨⟨b⟩⟩) be anisotropic. Then φ = φ_{1} + φ_{2} in W (F) for some forms φ_{1} and φ_{2} over F satisfying:

(1) φ_{1} ∈ ⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(⟨⟨b⟩⟩) is anisotropic.

(2) φ_{2} ∈ ann_{W (F)}(⟨⟨b⟩⟩).

(3) (φ_{2})_{K} is anisotropic.

Proof. By Corollary 6.23 the dimension of φ is even. We induct on dim φ. If φ_{K} is hyperbolic, then φ = φ_{1} works by Corollary 34.12 and if φ_{K} is anisotropic, then φ = φ_{2} works. So we may assume that φ_{K} is isotropic but not hyperbolic. In particular, dim φ ≥ 4. By Proposition 34.8, we can write

φ ≃ x⟨⟨a⟩⟩ ⊥ μ

for some x ∈ F^{×} and even-dimensional form μ over F. As φ ∈ ann_{W (F)}(⟨⟨b⟩⟩), we have ⟨⟨b⟩⟩ · μ = −x⟨⟨b, a⟩⟩ in W (F), so dim(⟨⟨b⟩⟩ ⊗ μ)_{an} = 0 or 4. Therefore, by Proposition 6.25, we can write

μ ≃ μ_{1} ⊥ y⟨⟨c⟩⟩

for some y, c ∈ F^{×} and even-dimensional form μ_{1} ∈ ann_{W (F)}(⟨⟨b⟩⟩). Substituting in the previous isometry and taking determinants, we see that ac ∈ D(⟨⟨b⟩⟩) by Proposition 6.25. Thus c = az for some z ∈ D(⟨⟨b⟩⟩). Consequently,

φ ≃ x⟨⟨a⟩⟩ ⊥ y⟨⟨az⟩⟩ ⊥ μ_{1} = x⟨⟨a, −xyz⟩⟩ + y⟨⟨z⟩⟩ + μ_{1}

in W (F). Let μ_{2} ≃ (y⟨⟨z⟩⟩ ⊥ μ_{1})_{an}. As y⟨⟨z⟩⟩ lies in ann_{W (F)}(⟨⟨b⟩⟩), so does μ_{2} and hence also x⟨⟨a, −xyz⟩⟩. By induction on dim φ, we can write μ_{2} = φ̃_{1} + φ̃_{2} in W (F) where φ̃_{1} satisfies condition (1) and φ̃_{2} satisfies conditions (2) and (3). It follows that

φ_{1} ≃ (⟨⟨a, −xyz⟩⟩ ⊥ φ̃_{1})_{an} and φ_{2} ≃ φ̃_{2}

work. □

Exercise 37.2. Let φ and ψ be 2-fold Pfister forms over a field of characteristic not 2. Prove that the group φW (F) ∩ ann_{W (F)}(ψ) ∩ I^{2}(F) is generated by 2-fold Pfister forms ρ in ann_{W (F)}(ψ) that are divisible by φ. This exercise generalizes. (Cf. Exercise 41.8 below.)

As we are interested in the case of fields of characteristic not 2, we shall show that to test finiteness of the u-invariant, it suffices to look at ann_{W (F)}(2⟨1⟩). For a field of characteristic not 2, let

u′(F) := max { dim φ | φ is an anisotropic form over F and 2φ = 0 in W (F)}

or ∞ if no such maximum exists.

Lemma 37.3. Let F be a field of characteristic not 2. Then u′(F) is finite if and only if u(F) is finite. Moreover, if u(F) is finite, then u(F) = u′(F) = 0 or u′(F) ≤ u(F) < 2u′(F).

Proof. We may assume that u′(F) > 0, i.e., that F is not a formally real pythagorean field. Let φ be an n-dimensional anisotropic form over F. Suppose that n ≥ 2u′(F). By Proposition 6.25, we can write φ ≃ μ_{1} ⊥ φ_{1} with μ_{1} ∈ ann_{W (F)}(2⟨1⟩) and 2φ_{1} anisotropic. By assumption, dim μ_{1} ≤ u′(F). Thus

2u′(F) ≤ dim φ = dim μ_{1} + dim φ_{1} ≤ u′(F) + dim φ_{1},

hence 2u′(F) ≤ dim 2φ_{1}. As (2φ)_{an} ≃ 2φ_{1}, we have dim(2φ)_{an} ≥ 2u′(F). Repeating the argument, we see inductively that dim(2^{m}φ)_{an} ≥ 2u′(F) for all m. In particular, φ is not torsion. The result follows. □

Hoffmann has shown that there exist fields F satisfying u′(F) < u(F). (Cf. [54].)

Let K/F be a quadratic extension. As it is not true that u(F) is finite if and only if u(K) is when F is formally real, we need a further condition for this to be true. This condition is given by a relative u-invariant.

Let F be a field of characteristic not 2 and L/F a field extension. The relative u-invariant of L/F is defined as

u(L/F) := max {dim(φ_{L})_{an} | φ a quadratic form over F

with φ_{L} torsion in W (L)}

or ∞ if no such integer exists.

The introduction of this invariant and its study come from Lee in [91] and [40]. We shall prove

Theorem 37.4. Let F be a field of characteristic different than 2 and K a quadratic extension of F. Then u(F) and u(K/F) are both finite if and only if u(K) is finite. Moreover, we have:

(1) If u(F) and u(K/F) are both finite, then u(K) ≤ u(F) + u(K/F). If, in addition, K is not formally real, then u(K) ≤ \frac{1}{2}u(F) + u(K/F).

(2) If u(K) is finite, then u(K/F) ≤ u(K) and u(F) < 6u(K) or u(F) = u(K) = 0. If, in addition, K is not formally real, then u(F) < 4u(K).

Proof. Let K = F (\sqrt{a}) and s_{∗} : W (K) → W (F) be the transfer induced by the F -linear functional defined by s(1) = 0 and s(\sqrt{a}) = 1.

Claim: Let φ be an anisotropic quadratic form over K such that s_{∗}(φ) is torsion in W (F). Then there exist a form σ over F and a form ψ over K satisfying:

(a) dim σ = dim φ.

(b) ψ is a torsion form in W (K).

(c) dim ψ ≤ 2 dim φ and φ ≃ (σ_{K} ⊥ ψ)_{an}.

(d) If s_{∗}(φ) is anisotropic over F, then dim φ ≤ dim ψ.

In particular, if u(F) is finite and s_{∗}(φ) anisotropic and torsion, then dim φ ≤ \frac{1}{2}u(F) and dim ψ ≤ u(F).

Let 2^{n}s_{∗}(φ) = 0 in W (F) for some integer n. By Corollary 34.3 with ρ = 2^{n}⟨1⟩, there exists a form σ over F such that dim σ = dim φ and 2^{n}φ ≃ 2^{n}σ_{K}. Let ψ ≃ (φ ⊥ (−σ))_{an}. Then ψ is a torsion form in W (K) as it has trivial total signature. Condition (c) holds by construction and (d) holds as s_{∗}(ψ) = s_{∗}(φ) in W (F).

We now prove (1). Suppose that both u(F) and u(K/F) are finite. Let τ be an anisotropic torsion form over K. By Proposition 34.1, there exists an isometry τ ≃ φ ⊥ μ_{K} for some form τ over K satisfying s_{∗}(φ) is anisotropic and some form μ over F. As s_{∗}(φ) = s_{∗}(τ) is torsion, we can apply the claim to φ. Let σ over F and ψ over K be forms as in the claim. By the last statement of the claim, we have dim φ ≤ \frac{1}{2}u(F). In particular, we have dim ψ ≤ 2 dim φ ≤ u(F) and φ = ψ + σ_{K} in W (K). Since τ and ψ are torsion, so is (σ + μ)_{K}. As τ = ψ + ((σ ⊥ μ)_{K})_{an} in W (K), it follows that dim τ ≤ u(F) + u(K/F) as needed.

Finally, if K is not formally real, then as above, we have τ ≃ φ ⊥ μ_{K} with dim φ ≤ \frac{1}{2}u(F). As every F -form is torsion in W (K), we have dim μ_{K} ≤ u(K/F) and the proof of (1) is complete.

We now prove (2). Suppose that u(K) is finite. Certainly u(K/F) ≤ u(K). We show the rest of the first statement. By Lemma 37.3, it suffices to show that u′(F) ≤ 3u′(K). Let φ ∈ ann_{W (F)}(2⟨1⟩) be anisotropic. By Lemma 37.1 and Corollary 34.33, we can decompose φ = φ_{1} +φ_{2} in W (F) with φ_{2} ∈ ann_{W (F)}(2⟨1⟩) satisfying (φ_{2})_{K} is anisotropic and φ_{1} is anisotropic over F and lies in

⟨⟨a⟩⟩W (F) ∩ ann_{W (F)}(2⟨1⟩) ⊂ ann_{W (F)}(⟨⟨a⟩⟩) ∩ ann_{W (F)}(2⟨1⟩)

using Lemma 34.33. In particular, (φ_{2})_{K} ∈ ann_{W (K)}(2⟨1⟩) so dim φ_{2} ≤ u′(K). Consequently, to show that u′(F) ≤ 3u′(K), it suffices to show dim φ_{1} ≤ 2u′(K). This follows from (i) of the following (with σ = φ_{1}):

Claim: Let σ be a nondegenerate quadratic form over F.

(i) If σ ∈ ann_{W (F)}(⟨⟨a⟩⟩) ∩ ann_{W (F)}(2⟨1⟩), then dim σ_{an} ≤ 2u′(K).

(ii) If σ ∈ ann_{W (F)}(⟨⟨a⟩⟩) ∩ W_{t}(F), then dim σ_{an} ≤ 2u(K) with inequality if K is not formally real.

By Corollary 34.33, in the situation of (i), there exists τ ∈ ann_{W (K)}(2⟨1⟩) such that σ = s_{∗}(τ). Then dim σ_{an} ≤ dim s_{∗}(τ_{an}) ≤ 2 dim τ_{an} ≤ 2u′(K) as needed.

We turn to the proof of (ii) which implies the the bound on u(F) in statement (2) for arbitrary K (with σ = φ_{1}). In the situation of (ii), we have dim σ_{an} ≤ u(K) by Corollaries 34.12 and 34.32. If K is not formally real, then any u(K)-dimensional form τ over K is universal. In particular, D(τ) ∩ F^{×} ≠ ∅ and (ii) follows.

Now assume that K is not formally real. Let φ be an anisotropic torsion form over F of dimension u(F). As im(s_{∗}) = ann_{W (F)}(⟨⟨a⟩⟩) by Corollary 34.12, using Proposition 6.25, we have a decomposition φ ≃ φ_{3} ⊥ φ_{4} with φ_{4} a form over F satisfying ⟨⟨a⟩⟩ ⊗ φ_{4} is anisotropic and φ_{3} ≃ s_{∗}(τ) for some form τ over K. Since φ_{3} lies in

s_{∗}(W (K)) = s_{∗}(W_{t}(K)) = ann_{W (F)}(⟨⟨a⟩⟩) ∩ W_{t}(F)

by Corollaries 34.12 and 34.32, we have dim φ_{3} < 2u(K) by the claim above. As ⟨⟨a⟩⟩·φ_{4} = ⟨⟨a⟩⟩·φ in W (F), hence is torsion, we have dim φ_{4} ≤ u(F)/2. Therefore, 2u(K) > dim φ_{3} = dim φ − dim φ_{4} ≥ u(F) − u(F)/2 and u(F) < 4u(K). □

Of course, by Theorem 36.6 if F is not formally real and K = F (\sqrt{a}) is a quadratic extension, then u(K) ≤ \frac{3}{2}u(F).

Corollary 37.5. Let F be a field of transcendence degree n over a real closed field. Then u(F) < 2^{n+2}.

Proof. F (\sqrt{−1}) is a C_{n}-field by Corollary 97.7. □

Remark 37.6. Let F be a field of characteristic not 2 and K/F a finite normal extension. Suppose that u(K) is finite. If K/F is quadratic, then the proof of Theorem 37.4 shows that u′(F) ≤ 3u′(K). If K/F is of degree 2^{r}m with m odd, arguing as in Proposition 36.13, shows that u′(F) ≤ 3^{r}u′(K), hence u(F) ≤ 2 · 3^{r}u(K).

One case where the bound in the remark can be sharpened is the following which generalizes the case of a pythagorean field of characteristic different from 2.

Proposition 37.7. Let F be a field of characteristic different from 2 and K/F a finite normal extension. If u(K) ≤ 2, then u(F) ≤ 2.

Proof. By Proposition 35.1, we know for a field E that I^{2}(E) is torsion-free if and only if E satisfies A_{2}, i.e., there are no anisotropic 2-fold torsion Pfister forms. In particular, as u(K) ≤ 2, we have I^{2}(K) is torsion-free. Arguing as in Proposition 36.13, we reduce to the case that K = F (\sqrt{a}) is a quadratic extension of F, hence I^{2}(F) is also torsion-free by Theorem 35.12. It follows that every torsion element ρ in I(F) lies in ann_{W (F)}(2⟨1⟩). In particular, by Proposition 6.25, we can write ρ ≃ ⟨⟨w⟩⟩ mod I^{2}(F) for some w ∈ D(2⟨1⟩), hence ρ ≃ ⟨⟨w⟩⟩ some w ∈ D(2⟨1⟩) and is universal. Consequently, every even-dimensional anisotropic torsion form over F is of dimension at most two. Suppose that there exists an odd-dimensional anisotropic torsion form φ over F. Then F is not formally real, hence all forms are torsion. As every 2-dimensional form over F is universal by the above, we must have dim φ = 1. The result follows. □

Corollary 37.8. Let F be a field of transcendence degree one over a real closed field. Then u(F) ≤ 2.

Exercise 37.9. Let F be a field of arbitrary characteristic and a ∈ F^{×} totally positive. If K = F (\sqrt{a}), then u(K) ≤ 2u(F).

We next show if K/F is a quadratic extension with K not formally real, then the relative u-invariant already determines finiteness. We note

Remark 37.10. Suppose that char F ≠ 2 and K = F (\sqrt{a}) is a quadratic extension of F that is not formally real. If φ is a nondegenerate quadratic form over F, then, by Proposition 34.8, there exist forms φ_{1} and ψ such that φ ≃ ⟨⟨a⟩⟩ ⊗ ψ ⊥ φ_{1} with dim φ_{1} ≤ u(K/F).

We need the following simple lemma.

Lemma 37.11. Let F be a field of characteristic different from 2 and K = F (\sqrt{a}) a quadratic extension of F that is not formally real. Suppose that u(K/F) < 2^{m}. Then I^{m+1}(F) is torsion-free, I^{m+1}(K) = 0, and the exponent of W_{t}(F) is at most 2^{m+1}.

Proof. If ρ ∈ P_{m}(F), then r_{K/F}^{∗} (ρ) = 0 as K is not formally real. So I^{m}(F) = ⟨⟨a⟩⟩I^{m−1}(F) by Theorem 34.22. It follows that I^{m+1}(K) = 0 by Lemma 34.16. Hence I^{m+1}(F (\sqrt{−1})) = 0 by Corollary 35.14. The result follows by Corollary 35.27. □

If F is a local field in the above, then one can show that u(K/F) = 2 for any quadratic extension K of F, but neither I^{2}(F) nor I^{2}(K) is torsion-free.

Theorem 37.12. Let F be a field of characteristic different from 2. Suppose that K is a quadratic extension of F with K not formally real. Then u(K/F) is finite if and only if u(K) is finite.

Proof. By Theorem 37.4, we may assume that u(K/F) is finite and must show that u(F) is also finite. Let φ be an anisotropic form over F satisfying 2φ = 0 in W (F). By the lemma, I^{n−1}(F) is torsion-free for some n ≥ 1. We apply the Remark 37.10 iteratively. In particular, if dim φ is large, then φ ≃ xρ ⊥ ψ for some ρ ∈ P_{n}(F) (cf. the proof of Proposition 36.12). Indeed, computation shows that if u(K/F) < 2^{m} and dim φ > 2^{m}(2^{m+2} − 1), then n = m + 2 works. As ρ is an anisotropic Pfister form and I^{n}(F) is torsion-free, 2ρ is also anisotropic. Scaling φ, we may assume that x = 1. Write ψ ≃ φ_{1} ⊥ φ_{2} with 2φ_{1} = 0 in W (F) and 2φ_{2} anisotropic. Then we have 2ρ ≃ 2(−φ_{2}). If b ∈ D(−φ_{2}), then 2⟨⟨b⟩⟩ · ρ is isotropic, hence is zero in W (F). As I^{n}(F) is torsion-free, ⟨⟨b⟩⟩ · ρ = 0 in W (F) and b ∈ D(ρ). It follows that φ cannot be anisotropic if dim φ > 2^{m}(2^{m+2} − 1). By Lemma 37.3, it follows that u(F) ≤ 2^{m+1}(2^{m+2} − 1) and the result follows by Theorem 37.4. □

The bounds in the proof can be improved but are still very weak. The theorem does not generalize to the case when K is formally real. Indeed, let F_{0} be a formally real subfield of the algebraic closure of the rationals having square classes represented by ±1, ±w where w is a sum of (two) squares. Let F = F_{0}((t_{1}))((t_{2})) ⋯ and K = F (\sqrt{w}). Then, using Corollary 34.12, we see that u(K/F) = 0 but both u(F) and u(K) are infinite.

Corollary 37.13. Let F be a field of characteristic different from 2. Then u(K) is finite for all finite extensions of F if and only if u(F (\sqrt{−1})) is finite if and only if u(F (\sqrt{−1})/F) is finite.

Remark 37.14. Using the theory of abstract Witt rings, Schubert proved that if F is a formally real field, then u(K) is finite for all finite extensions of F if and only if u(F) is finite and I_{red}^{n+1}(F) = 2I_{red}^{n}(F) for some n if and only if u(F (\sqrt{−1})) is finite (cf. [122]). Recall that the condition on the reduced Witt ring is equivalent to the cokernel of the total signature map having finite exponent by Proposition 35.23.

38. Construction of fields with even u-invariant

In 1953 Kaplansky conjectured in [68] that u(F) if finite was always a power of two. This was shown to be false in [101] where it was shown there exist fields having u-invariant six and afterwards in [102] having u-invariant any even integer. Subsequently, Izhboldin constructed fields having u-invariant 9 in [64] and Vishik has constructed fields having u-invariant 2^{r} + 1 for every r ≥ 3 in [135].

By taking iterated Laurent series fields over the complex numbers, we can construct fields whose u-invariant is 2^{n} for any n ≥ 0. (We also know that formally real pythagorean fields have u-invariant zero.) In this section, given any even integer m > 0, we construct fields whose u-invariant is m. This construction was first done in [102].

Lemma 38.1. Let φ ∈ I_{q}^{2}(F) be a form of dimension 2n ≥ 2. Then φ is a sum of n − 1 general quadratic 2-fold Pfister forms in I_{q}^{2}(F) and ind clif(φ) ≤ 2^{n−1}.

Proof. We induct on n. If n = 1, we have φ = 0 and the statement is clear. If n = 2, φ is a general 2-fold Pfister form and by Proposition 12.4, we have clif(φ) = [Q], where Q is a quaternion algebra such that Nrd_{Q} is similar to φ. Hence ind clif(φ) ≤ 2.

In the case n ≥ 3 write φ = σ ⊥ ψ where σ is a binary form. Choose a ∈ F^{×} such that the form aσ ⊥ ψ is isotropic, i.e., aσ ⊥ ψ ≃ ℍ ⊥ μ for some form μ of dimension 2n − 2. We have in I_{q}(F):

φ = σ + ψ = ⟨⟨a⟩⟩σ + μ

and therefore clif(φ) = clif(⟨⟨a⟩⟩σ) · clif(μ) by Lemma 14.2. Applying the induction hypothesis to μ, we have φ is a sum of n − 1 general quadratic 2-fold Pfister forms and

ind clif(φ) ≤ ind clif(⟨⟨a⟩⟩σ) · ind clif(μ) ≤ 2 · 2^{n−2} = 2^{n−1}.  □

Corollary 38.2. In the condition of the lemma assume that ind clif(φ) = 2^{n−1}. Then φ is anisotropic.

38. CONSTRUCTION OF FIELDS WITH EVEN u-INVARIANT 171

Proof. Suppose φ is isotropic, i.e., φ ≃ ℍ ⊥ ψ for some ψ of dimension 2n − 2. Applying Lemma 38.1 to ψ, we have ind clif(φ) = ind clif(ψ) ≤ 2^{n−2}, a contradiction. □

Lemma 38.3. Let D be a tensor product of n − 1 quaternion algebras (n ≥ 1). Then there is a φ ∈ I_{q}^{2}(F) of dimension 2n such that clif(φ) = [D] in Br(F).

Proof. We induct on n. The case n = 1 follows from Proposition 12.4. If n ≥ 2 write D = Q ⊗ B, where Q is a quaternion algebra and B is a tensor product of n − 2 quaternion algebras. By the induction hypothesis, there is ψ ∈ I_{q}^{2}(F) of dimension 2n − 2 such that clif(ψ) = [B]. Choose a quadratic 2-fold Pfister form σ with clif(σ) = [Q] and an element a ∈ F^{×} such that aσ ⊥ ψ is isotropic, i.e., aσ ⊥ ψ ≃ ℍ ⊥ φ for some φ of dimension 2n. Then φ works as clif(φ) = clif(σ) · clif(ψ) = [Q] · [B] = [D]. □

Let 𝔄 be a set (of isometry classes) of irreducible quadratic forms. For any finite subset S ⊂ 𝔄 let X_{S} be the product of all the quadrics X_{φ} with φ ∈ S. If S ⊂ T are two subsets of 𝔄 we have the dominant projection X_{T} → X_{S} and therefore the inclusion of function fields F (X_{S}) → F (X_{T}). Set F_{𝔄} = colim F_{S} over all finite subsets S ⊂ 𝔄. By construction, all quadratic forms φ ∈ 𝔄 are isotropic over the field extension F_{𝔄}/F. A field is called 2-special if every finite extension of it has degree a power of 2 (cf. §101.B).

Theorem 38.4. Let F be a field and n ≥ 1 an integer. Then there is a field extension E of F satisfying:

(1) u(E) = 2n.

(2) I_{q}^{3}(E) = 0.

(3) E is 2-special.

Proof. To every field L, we associate three fields L^{(1)}, L^{(2)}, and L^{(3)} as follows: Let 𝔄 be the set of isometry classes of all nondegenerate quadratic forms over

L of dimension 2n + 1. We set L^{(1)} = L_{𝔄}. Every nondegenerate quadratic form over L of dimension 2n + 1 is isotropic over L^{(1)}.

Let 𝔅 be the set of isometry classes of all quadratic 3-fold Pfister forms over L. We set L^{(2)} = L_{𝔅}. By construction, every quadratic 3-fold Pfister form over L is isotropic over L^{(2)}.

Finally, let L^{(3)} be a 2-special closure of L (cf. §101.B).

Let D be a central division L-algebra of degree 2^{n−1}. By Corollaries 30.9, 30.11, and 98.5, the algebra D remains a division algebra when extended to L^{(1)}, L^{(2)}, or L^{(3)}.

Let L be a field extension of F such that there is a central division algebra D over L that is a tensor product of n−1 quaternion algebras (cf. Proposition 98.18). By Lemma 38.3, there is a φ ∈ I_{q}^{2}(L) of dimension 2n such that clif(φ) = [D] in Br(L).

We construct a tower of field extensions E_{0} ⊂ E_{1} ⊂ E_{2} ⊂ … by induction. We set E_{0} = L. If E_{i} is defined we set E_{i+1} = (((E_{i})^{(1)})^{(2)})^{(3)}. Note that the field E_{i+1} is 2-special and all nondegenerate quadratic forms of dimension 2n + 1 and all 3-fold Pfister forms over E_{i} are isotropic over E_{i+1}. Moreover, the algebra D remains a division algebra over E_{i+1}.

Now set E = ⋃ E_{i}. Clearly, E has the following properties:

(i) All (2n + 1)-dimensional Pfister forms over E are isotropic. In particular, u(E) ≤ 2n.

(ii) The field E is 2-special.

(iii) All quadratic 3-fold Pfister forms over E are isotropic. In particular,

I_{q}^{3}(E) = 0 . (iv) The algebra D_{E} is a division algebra.

As clif(φ_{E}) = [D_{E}], it follows from Corollary 38.2 that φ_{E} is anisotropic. In particular, u(E) = 2n. □

39. Addendum: Linked fields and the Hasse number

Theorem 39.1. Let F be a field. Then the following conditions are equivalent:

(1) Every pair of quadratic 2-fold Pfister forms over F are linked.

(2) Every 6-dimensional form in I_{q}^{2}(F) is isotropic.

(3) The tensor product of two quaternion algebras over F is not a division algebra.

(4) Every two division quaternion algebras over F have isomorphic separable quadratic subfields.

(5) Every two division quaternion algebras over F have isomorphic quadratic subfields.

(6) The classes of quaternion algebras in Br(F) form a subgroup.

Proof. (1) ⇒ (2): Let ψ be a 6-dimensional form in I_{q}^{2}(F). By Lemma 38.1, we have ψ = φ_{1}+φ_{2}, where φ_{1} and φ_{2} are general quadratic 2-fold Pfister forms. By assumption, φ_{1} and φ_{2} are linked. Therefore, the class of ψ in I_{q}^{2}(F) is represented by a form of dimension 4, hence ψ is isotropic.

(2) ⇒ (1): Let φ_{1} and φ_{2} be two quadratic 2-fold Pfister forms over F. Then φ_{1} − φ_{2} = ψ for some 6-dimensional form ψ ∈ I_{q}^{2}(F). As ψ is isotropic, we have 𝔦_{0}(φ_{1} ⊥ −φ_{2}) ≥ 2, i.e., φ_{1} and φ_{2} are linked.

(1) ⇒ (4): Let Q_{1} and Q_{2} be division quaternion algebras over F. Let φ_{1} and φ_{2} be the reduced norm quadratic forms of Q_{1} and Q_{2}, respectively. By assumption, φ_{1} and φ_{2} are linked. In particular, φ_{1} and φ_{2} are split by a separable quadratic field extension L/F. Hence L splits Q_{1} and Q_{2} and therefore L is isomorphic to subfields of Q_{1} and Q_{2}.

(3) ⇔ (4) ⇔ (5) is proven in Theorem 98.19.

(3) ⇔ (6) is obvious.

(4) ⇒ (1): Let φ_{1} and φ_{2} be two anisotropic 2-fold Pfister forms over F. Let Q_{1} and Q_{2} be two division quaternion algebras with the reduced norm forms φ_{1} and φ_{2}, respectively. By assumption, Q_{1} and Q_{2} have quadratic subfields isomorphic to a separable quadratic extension L/F. By Example 9.7, the forms φ_{1} and φ_{2} are divisible by the norm form of L/F and hence are linked. □

A field F is called linked if F satisfies the conditions of Theorem 39.1.

For a formally real field F, the u-invariant can be thought of as a weak Hasse Principle, i.e., every locally hyperbolic form of dimension > u(F) is isotropic. A variant of the u-invariant introduced in [31] naturally arises. We call a quadratic form φ over F locally isotropic or totally indefinite if φ_{F_{P}} is isotropic at each real closure F_{P} of F (if any), i.e., φ is indefinite at each real closure of F (if any). The Hasse number of a field F is defined to be

ũ(F) := max { dim φ | φ is a locally isotropic anisotropic form over F }

or ∞ if no such maximum exists. For fields that are not formally real this coincides with the ū-invariant. If a field is formally real, finiteness of its ũ-invariant is a very strong condition and is a form of a strong Hasse Principle. For example, if F is a global field, then ũ(F) = 4 by Meyer’s Theorem [104] (a forerunner of the Hasse-Minkowski Principle [121]) and if F is the function field of a real curve, then ũ(F) = 2 (cf. Example 39.11 below), but if F/ℝ is formally real and finitely generated of transcendence degree > 1, then, although u(F) is finite, its Hasse number ũ(F) is infinite.

Exercise 39.2. Show, if the Hasse number is finite, then it cannot be 3, 5, or 7.

We establish another characterization of ũ(F) using a concept introduced in [38]. We say F satisfies Property H_{n} with n > 1 if there exist no anisotropic, locally isotropic forms of dimension n. Thus if ũ(F) is finite, then

ũ(F) + 1 = min {n | F satisfies H_{m} for all m ≥ n}.

Remark 39.3. Every 6-dimensional form in I_{q}^{2}(F) is locally isotropic, since every element in I^{2}(F) has signature divisible by 4 at every ordering. Hence if ũ(F) ≤ 4, then F is linked by Theorem 39.1.

Lemma 39.4. Let F be a linked field of characteristic not 2. Then (1) Any pair of n-fold Pfister forms are linked for n ≥ 2.

(2) If φ ∈ P_{n}(F), then φ ≃ ⟨⟨−w_{1}, x⟩⟩ if n = 2 and φ = 2^{n−3}⟨⟨−w_{1}, −w_{2}, x⟩⟩ if n ≥ 3 for some w_{1}, w_{2} ∈ D(3⟨1⟩) and x ∈ F^{×}.

(3) For every n ≥ 0 and φ ∈ I^{n}(F), there exists an integer m and ρ_{i} ∈ GP_{i}(F) with n ≤ i ≤ m satisfying φ = ∑_{i=n}^{m} ρ_{i} in W (F). Moreover, if φ is a torsion element, then each ρ_{i} is torsion.

(4) I^{4}(F) is torsion-free.

Proof. (1), (2): Any pair of n-fold Pfister forms is easily seen to be linked by induction, so (1) is true. As any 2-fold Pfister form is linked to 4⟨1⟩, statement (2) holds for n = 2. Let ρ = ⟨⟨a, b, c⟩⟩ be a 3-fold Pfister form, then applying the n = 2 case gives ρ = ⟨⟨w_{1}, x, y⟩⟩ = ⟨⟨w_{1}, w_{2}, z⟩⟩ for some x, y, z ∈ F^{×} and w_{1}, w_{2} ∈ D(3⟨1⟩). This establishes the n = 3 case. Let ρ = ⟨⟨a, b, c, d⟩⟩ be a 4-fold Pfister form. By assumption, there exist x, y, z ∈ F^{×} such that ⟨⟨a, b⟩⟩ ≃ ⟨⟨x, y⟩⟩ and ⟨⟨c, d⟩⟩ ≃ ⟨⟨x, z⟩⟩. Thus

(39.5)  ρ = ⟨⟨a, b, c, d⟩⟩ ≃ ⟨⟨x, y, x, z⟩⟩ ≃ ⟨⟨−1, y, x, z⟩⟩ = 2⟨⟨y, x, z⟩⟩.

Statement (2) follows.

(3): Let ψ and τ be n-fold Pfister forms. As they are linked, ψ − τ = a⟨⟨b⟩⟩ · μ in W (F) for some (n − 1)-fold Pfister form μ and a, b ∈ F^{×}. Then

xψ + yτ = xψ − xτ + xτ + yτ = ax⟨⟨b⟩⟩ · μ + x⟨⟨−xy⟩⟩ · τ.

The first part now follows by repeating this argument. If φ is torsion, then inductively, each ρ_{i} is torsion by the Hauptsatz 23.7, so the second statement follows.

(4): By (3), it suffices to show there are no anisotropic torsion n-Pfister forms with n > 3. By Proposition 35.3, it suffices to show if ρ ∈ P_{4}(F) satisfies 2ρ = 0 in W (F), then ρ = 0 in W (F). By Lemma 35.2, we can write ρ ≃ ⟨⟨a, b, c, w⟩⟩ with w ∈ D(2⟨1⟩) and a, b, c ∈ F^{×}. Applying equation (39.5) with d = w, we have ρ ≃ 2⟨⟨y, x, z⟩⟩ ≃ 2⟨⟨y, c, w⟩⟩ which is hyperbolic. The result follows. □

Lemma 39.6. Let char F ≠ 2 and n ≥ 2. If F is linked and F satisfies H_{n}, then it satisfies H_{n+1}.

Proof. Let φ be an (n+1)-dimensional anisotropic quadratic form with n ≥ 2. Replacing φ by xφ for an appropriate x ∈ F^{×}, we may assume that φ = ⟨w, b, wb⟩ ⊥ φ_{1} for some w, b ∈ F^{×} and a form φ_{1} over F and by Lemma 39.4 that w ∈ D(3⟨1⟩). Let φ_{2} = ⟨w, b⟩ ⊥ φ_{1}. As sgn_{P} ⟨b⟩ = sgn_{P} ⟨wb⟩ for all P ∈ 𝔛(F), the form φ is locally isotropic if and only if φ_{2} is. The result follows by induction. □

Remark 39.7. If char F ≠ 2 and n ≥ 4, then F satisfies Property H_{n+1} if it satisfies Property H_{n}. However, in general, H_{3} does not imply H_{4} (cf. [38]).

Exercise 39.8. Let F be a formally real pythagorean field. Then ũ(F) is finite if and only if I^{2}(F) = 2I(F). Moreover, if this is the case, then ũ(F) = 0.

The following theorem from [31] strengthens the result in [35].

Theorem 39.9. Let F be a linked field of characteristic not 2. Then u(F) = ũ(F) and ũ(F) = 0, 1, 2, 4, or 8.

Proof. We first show that ũ(F) = 0, 1, 2, 4, or 8. We know that I^{4}(F) is torsion-free by Lemma 39.4. We first show that F satisfies H_{9}, hence ũ(F) ≤ 8 by Lemma 39.6. Let φ be a 9-dimensional locally isotropic form over F. Replacing φ by xφ for an appropriate x ∈ F^{×}, we can assume that φ = ⟨1⟩ + φ_{1} in W (F) with φ_{1} ∈ I^{2}(F) using Proposition 4.13. By Lemma 39.4, we have a congruence

(39.10)  φ ≡ ⟨1⟩ + ρ_{2} − ρ_{3} mod I^{4}(F)

for some ρ_{i} ∈ P_{i}(F) with i = 2, 3. Write ρ_{2} ≃ ⟨⟨a, b⟩⟩ and ρ_{3} ≃ ⟨⟨c, d, e⟩⟩. As F is linked, we may assume that e = b and −d ∈ D_{F} (ρ′_{2}). Thus we have

φ ≡ ⟨1⟩ + ⟨⟨a, b⟩⟩ − ⟨⟨c, d, b⟩⟩
≡ ⟨1⟩ − d(⟨⟨a, b⟩⟩ − ⟨⟨c, b⟩⟩) − ⟨⟨c, b⟩⟩
≡ −cd⟨⟨ac, b⟩⟩ − ⟨⟨c, b⟩⟩′ mod I^{4}(F).

Let μ = φ ⊥ cd⟨⟨ac, b⟩⟩ ⊥ ⟨⟨c, b⟩⟩′ be a locally isotropic form over F lying in I^{4}(F). In particular, for all P ∈ 𝔛(F), we have 16 | sgn_{P} μ. As the locally isotropic form μ is 16-dimensional, | sgn_{P} μ| < 16 for all P ∈ 𝔛(F), so sgn_{P} μ = 0 for all P ∈ 𝔛(F) and μ ∈ I_{t}^{4}(F) = 0. Consequently, φ = −cd⟨⟨ac, b⟩⟩ ⊥ (−⟨⟨c, b⟩⟩′) in W (F), so φ is isotropic and ũ(F) ≤ 8.

Suppose that ũ(F) < 8. Then there are no anisotropic torsion 3-fold Pfister forms over F. It follows that I^{3}(F) is torsion-free by Lemma 39.4. We show ũ(F) ≤ 4. To do this it suffices to show that F satisfies H_{5} by Lemma 39.6. Let φ be a 5-dimensional, locally isotropic space over F. Arguing as above but going mod I^{3}(F), we may assume that

φ ≡ ⟨1⟩ − ⟨⟨a, b⟩⟩ = −⟨⟨a, b⟩⟩′ mod I^{3}(F).

Let μ = φ ⊥ ⟨⟨a, b⟩⟩′ be an 8-dimensional, locally isotropic form over F lying in I^{3}(F). As above, it follows that μ is locally hyperbolic, hence μ ∈ I_{t}^{3}(F) = 0. Thus φ = −⟨⟨a, b⟩⟩′ in W (F) so it is isotropic and ũ(F) < 4. In a similar way, we see that ũ(F) = 0, 1, 2 are the only other possibilities. This shows that ũ(F) = 0, 1, 2, 4, 8. The argument above and Lemma 39.4 show that u(F) = ũ(F). □

Note the proof shows that if F is linked and I^{n}(F) is torsion-free, then ũ(F) ≤ 2^{n−1}.

It is still unknown whether Theorem 39.9 is true if char F = 2.

Example 39.11. (1) If F (\sqrt{−1}) is a C_{1} field, then I^{2}(F (\sqrt{−1})) = 0. It follows that I^{2}(F) = 2I(F) and is torsion-free by Corollary 35.14 and Proposition 35.1 (or Corollary 35.27). In particular, F is linked and ũ(F) ≤ 2.

(2) If F is a local or global field, then ũ(F) = 4.

(3) Let F_{0} be a local field and F = F_{0}((t)) be a Laurent series field. As u(F_{0}) = 4, and F is not formally real, we have ũ(F) = u(F) = 8. If char F ≠ 2, this field F is linked by the following exercise:

Exercise 39.12. Let F be a field of characteristic not 2. Show that the field F ((t)) is linked if and only if there exist no 4-dimensional anisotropic quadratic forms over F_{0} of nontrivial discriminant.

There exist linked formally real fields with Hasse number 8, but the construction of such fields is more delicate (cf. [42]).

Remark 39.13. Let F be a formally real field. Then it can be shown that ũ(F) is finite if and only if u(F) is finite and I^{2}(F_{py}) = 2I^{n}(F_{py}) (cf. [42]). If both of these invariants are finite, they may be different (cf. [112].)

Remark 39.14. We say that a formally real field F satisfies condition S_{n} if the map s_{∗} : I_{red}^{n}(F (\sqrt{w})) → I_{red}^{n}(F) induced by s : F (\sqrt{w}) → F with s(1) = 0 for all w ∈ D(∞⟨1⟩) is surjective (cf. Exercise 35.4). In [42], it is shown that ũ(F) is finite if and only if u(F) is finite, F satisfies S_{1}, and st_{r}(F) ≤ 1. Moreover, it is also shown in [42] that st_{r}(F_{py}) ≤ n if and only if st_{r}(F) ≤ n and F satisfies S_{n}. This last statement is the real analog of the first three conditions of Corollary 35.27.

CHAPTER VII Applications of the Milnor Conjecture

40. Exact sequences for quadratic extensions

In this section, we derive the first consequences of the validity of the Milnor Conjecture for fields of characteristic different from 2. In particular, we show that the infinite complexes (34.20) and (34.21) of the powers of I and Ī arising from a quadratic extension of a field of characteristic different from 2 are in fact exact as shown in the paper [107] of Orlov, Vishik, and Voevodsky. For fields of characteristic 2, we also show this to be true for separable quadratic extensions as well as proving the exactness of the corresponding complexes (34.27) and (34.28) for purely inseparable quadratic extensions. In addition, we show that for all fields, the ideals I_{q}^{n}(F) coincide with the ideals J_{n}(F) defined in terms of the splitting patterns of quadratic forms.

We need the following lemmas.

Lemma 40.1. Let K/F be a quadratic field extension and s : K → F a nonzero F -linear functional. Then for every n ≥ 0, the diagram

k_{n}(K) ⟶^{c_{K/F}} k_{n}(F)
f_{n}↓  ↓^{f_{n}}
\overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F)

commutes where the vertical homomorphisms are defined in (5.1).

Proof. As all the maps in the diagram are K_{∗}(F)-linear, in view of Lemma 34.16, it is sufficient to check commutativity only when n = 1. The statement follows now from Corollary 34.19 and Fact 100.8. □

Lemma 40.2. Suppose that F is a field of characteristic 2 and K/F a quadratic field extension. Let s : K → F be a nonzero F -linear functional. Then the diagram

\overline{I}_{q}^{n}(K) ⟶^{s_{∗}} \overline{I}_{q}^{n}(F)
ē_{n}↓  ↓ē_{n}
H^{n}(K) ⟶^{c_{K/F}} H^{n}(F)

is commutative.

Proof. It follows from Lemma 34.16 that it is sufficient to prove the statement in the case n = 1. This follows from Lemmas 34.14 and 34.18, since the corestriction map c_{K/F} : H^{1}(K) → H^{1}(F) is induced by the trace map Tr_{K/F} : K → F (cf. Example 101.2). □

We set I^{n}(F) = W (F) if n ≤ 0.

We first consider the case char F ≠ 2.

Theorem 40.3. Suppose that F is a field of characteristic different from 2 and K = F (\sqrt{a})/F a quadratic extension with a ∈ F^{×}. Let s : K → F be an F -linear functional such that s(1) = 0. Then the following infinite sequences are exact:

⋯ ⟶^{s_{∗}} I^{n−1}(F) ⟶^{·⟨⟨a⟩⟩} I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} I^{n}(F) ⟶^{·⟨⟨a⟩⟩} I^{n+1}(F) → ⋯,

⋯ ⟶^{s_{∗}} \overline{I}^{n−1}(F) ⟶^{·⟨⟨a⟩⟩} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{·⟨⟨a⟩⟩} \overline{I}^{n+1}(F) → · · · .

Proof. Consider the diagram

k_{n−1}(F) ⟶^{·{a}} k_{n}(F) ⟶^{r_{K/F}} k_{n}(K) ⟶^{c_{K/F}} k_{n}(F) ⟶^{·{a}} k_{n+1}(F)
↓  ↓  ↓  ↓  ↓
\overline{I}^{n−1}(F) ⟶^{·⟨⟨a⟩⟩} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{·⟨⟨a⟩⟩} \overline{I}^{n+1}(F)

where the vertical homomorphisms are defined in (5.1). It follows from Lemma 40.1 that the diagram is commutative. By Fact 5.15, the vertical maps in the diagram are isomorphisms. The top sequence in the diagram is exact by Proposition 101.10. Therefore, the bottom sequence is also exact.

To prove exactness of the first sequence in the statement consider the commutative diagram

I^{n+1}(F) → I^{n+1}(K)→ I^{n+1}(F)→ I^{n+2}(F)→ I^{n+2}(K)
↓  ↓  ↓  ↓  ↓
I^{n−1}(F)→ I^{n}(F) → I^{n}(K) → I^{n}(F) → I^{n+1}(F)→ I^{n+1}(K)
↓  ↓  ↓  ↓  ↓
\overline{I}^{n−1}(F)→ \overline{I}^{n}(F) → \overline{I}^{n}(K) → \overline{I}^{n}(F) → \overline{I}^{n+1}(F)

with the horizontal sequences considered above and natural vertical maps. By the first part of the proof the bottom sequence is exact. Therefore, exactness of the middle sequence implies exactness of the top one. Thus the statement follows by induction on n (with the base of the induction given by Corollary 34.12). □

Remark 40.4. Let char F ≠ 2. Then the second exact sequence in Theorem 40.3 can be rewritten as

\xymatrix{ & GW (K) \ar[dr]^{s_{∗}} \\ GW (F) \ar[ur]^{r_{K/F}} &  & GW (F) \ar[ll]_{·⟨⟨a⟩⟩}}

is exact (cf. Corollary 34.12).

Now consider the case of fields of characteristic 2. We consider separately the cases of separable and purely inseparable quadratic field extensions.

Theorem 40.5. Suppose that F is a field of characteristic 2 and K/F a separable quadratic field extension. Let s : K → F be a nonzero F -linear functional such that s(1) = 0. Then the following sequences are exact:

0 → I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} I^{n}(F)

⟶^{· N_{K/F}} I_{q}^{n+1}(F) ⟶^{r_{K/F}} I_{q}^{n+1}(K) ⟶^{s_{∗}} I_{q}^{n+1}(F) → 0,

0 → \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F)

⟶^{· N_{K/F}} \overline{I}_{q}^{n+1}(F) ⟶^{r_{K/F}} \overline{I}_{q}^{n+1}(K) ⟶^{s_{∗}} \overline{I}_{q}^{n+1}(F) → 0.

Proof. Consider the diagram

0 → k_{n}(F) ⟶^{r_{K/F}} k_{n}(K) ⟶^{c_{K/F}} k_{n}(F)
↓  ↓  ↓
0 → \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F)
k_{n}(F) ⟶^{·[K]} H^{n+1}(F) ⟶^{r_{K/F}} H^{n+1}(K) ⟶^{c_{K/F}} H^{n+1}(F) → 0
↓  ↑  ↑  ↑
\overline{I}^{n}(F) ⟶^{· N_{K/F}} \overline{I}_{q}^{n+1}(F) ⟶^{r_{K/F}} \overline{I}_{q}^{n+1}(K) ⟶^{s_{∗}} \overline{I}_{q}^{n+1}(F) → 0

where the vertical homomorphisms are defined in (5.1) and Fact 16.2. The middle map in the top row is the multiplication by the class [K] ∈ H^{1}(F). By Proposition 101.12, the top sequence is exact. By Facts 5.15 and 16.2, the vertical maps are isomorphisms. Therefore, the bottom sequence is exact.

Exactness of the second sequence in the statement follows by induction on n from the first part of the proof and commutativity of the diagram

\begin{aligned}0 → I^{n+1}(F) & → I^{n+1}(K) & → I^{n+1}(F) & → I_{q}^{n+2}(F) & → I_{q}^{n+2}(K) & → I_{q}^{n+2}(F) → 0 \\ ↓ & ↓ & ↓ & ↓ & ↓ & ↓ \\ 0 → I^{n}(F) & → I^{n}(K) & → I^{n}(F) & → I_{q}^{n+1}(F) & → I_{q}^{n+1}(K) & → I_{q}^{n+1}(F) → 0 \\ ↓ & ↓ & ↓ & ↓ & ↓ & ↓ \\ 0 → \overline{I}^{n}(F) & → \overline{I}^{n}(K) & → \overline{I}^{n}(F) & → \overline{I}_{q}^{n+1}(F) & → \overline{I}_{q}^{n+1}(K) & → \overline{I}_{q}^{n+1}(F) → 0. \end{aligned}

The base of the induction follows from Corollary 34.15. □

Theorem 40.6. Suppose that F is a field of characteristic 2 and K/F a purely inseparable quadratic field extension. Let s : K → F be an F -linear functional such that s(1) = 0. Then the following sequences are exact:

⋯ ⟶^{s_{∗}} I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} I^{n}(F) ⟶^{r_{K/F}} I^{n}(K) ⟶^{s_{∗}} ⋯,

⋯ ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} · · · .

Proof. Consider the diagram

k_{n}(F) ⟶^{r_{K/F}} k_{n}(K) ⟶^{c_{K/F}} k_{n}(F) ⟶^{r_{K/F}} k_{n}(K)
↓  ↓  ↓  ↓
\overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K) ⟶^{s_{∗}} \overline{I}^{n}(F) ⟶^{r_{K/F}} \overline{I}^{n}(K)

where the vertical homomorphisms are defined in (5.1). The diagram is commutative by Lemma 40.1. By Fact 5.15, the vertical maps in the diagram are isomorphisms. The top sequence in the diagram is exact by Proposition 100.12. Therefore, the bottom sequence is also exact. The proof of exactness of the second sequence in the statement of the theorem is similar to the one in Theorems 40.3 and 40.5. □

For further applications, we shall also need the following result which follows immediately from the validity of the Milnor Conjecture proven by Voevodsky in [136] and Orlov, Vishik, and Voevodsky in [107, Th. 2.1]:

Fact 40.7. Let char F ≠ 2 and let ρ be a quadratic n-fold Pfister form over F. Then the sequence

∐ H^{∗}(L) \underrightarrow{∑ c_{L/F}} H^{∗}(F) ⟶^{∪e_{n}(ρ)} H^{∗+n}(F) ⟶^{r_{F (ρ)/F}} H^{∗+n}(F (ρ)),

is exact, where the direct sum is taken over all quadratic field extensions L/F such that ρ_{L} is isotropic.

Fact 40.8 (cf. [13, Th. 5.4]). Let char F = 2 and let ρ be a quadratic n-fold Pfister form over F. Then the kernel of r_{F (ρ)/F} : H^{n}(F) → H^{n}(F (ρ)) coincides with {0, e_{n}(ρ)}.

Corollary 40.9. Let ρ be a quadratic n-fold Pfister form over an arbitrary field F. Then the kernel of the natural homomorphism \overline{I}_{q}^{n}(F) → \overline{I}_{q}^{n}(F (ρ)) coincides with {0, ρ̄}.

Proof. Under the isomorphism \overline{I}_{q}^{n}(F) →^{∼} H^{n}(F) (cf. Fact 16.2), the homomorphism in the statement is identified with H^{n}(F) → H^{n}(F (ρ)). The statement now follows from Fact 40.7 if char F ≠ 2 and Fact 40.8 if char F = 2. □

The following statement generalizes Proposition 25.13 and was first proven by Orlov, Vishik, and Voevodsky in [107].

Theorem 40.10. If F is a field, then J_{n}(F) = I_{q}^{n}(F) for every n ≥ 1.

Proof. By Corollary 25.12, we have the inclusion I_{q}^{n}(F) ⊂ J_{n}(F). Let φ ∈ J_{n}(F). We show by induction on n that φ ∈ I_{q}^{n}(F). As φ ∈ J_{n−1}(F), by the induction hypothesis, we have φ ∈ I_{q}^{n−1}(F). The form φ is a sum of m general (n − 1)-fold Pfister forms in I_{q}^{n−1}(F) for some m. Let ρ be one of them. Let K = F (ρ). Since φ_{K} is a sum of m − 1 general (n − 1)-Pfister forms in I_{q}^{n−1}(K), by induction on m we have φ_{K} ∈ I_{q}^{n}(K). By Corollary 40.9, we have either φ ∈ I_{q}^{n}(F) or φ ≡ ρ modulo I_{q}^{n}(F). But the latter case does not occur as φ ∈ J_{n}(F) and ρ ∉ J_{n}(F). □

41. Annihilators of Pfister forms

The main purpose of this section is to establish the generalization of Corollary 6.23 and Theorem 9.12 on the annihilators of bilinear and quadratic Pfister forms and show these annihilators respect the grading induced by the fundamental ideal. We even show that if α is a bilinear or quadratic Pfister form, then the annihilator ann_{W (F)}(α) ∩ I^{n}(F) is not only generated by bilinear Pfister forms annihilated by α but is, in fact, generated by bilinear n-fold Pfister forms of the type 𝔟 ⊗ 𝔠 with 𝔟 ∈ ann_{W (F)}(α) a 1-fold bilinear Pfister form and 𝔠 a bilinear (n − 1)-fold Pfister form. In particular, Pfister forms of the type ⟨⟨w, a_{2}, …, a_{n}⟩⟩ with w ∈ D(∞⟨1⟩) and a_{i} ∈ F^{×} generate I_{t}^{n}(F), thus solving the problems raised at the end of §33. These results first appeared in [8].

Let F be a field. The smallest integer n such that I^{n+1}(F) = 2I^{n}(F) and I^{n+1}(F) is torsion-free is called the stable range of F and is denoted by st(F). We say that F has finite stable range if such an n exists and write st(F) = ∞ if such an n does not exist. By Corollary 35.30, a field F has finite stable range if and only if I^{n+1}(F) = 2I^{n}(F) for some integer n. If F is not formally real, then st(F) is the smallest integer n such that I^{n+1}(F) = 0. If F is formally real, then it follows from Corollary 35.27 that st(F) = st(F (\sqrt{−1})), i.e., st(F) is the smallest integer n such that I^{n+1}(F (\sqrt{−1})) = 0.

Lemma 41.1. Suppose that F has finite transcendence degree n over its prime subfield. Then st(F) ≤ n + 2 if char F = 0 and st(F) ≤ n + 1 if char F > 0.

Proof. If the characteristic of F is positive, then F is a C_{n+1}-field (cf. 97.7) as finite fields are C_{1} fields by the Chevalley-Warning Theorem (cf. [123, I.2, Th. 3]) and, therefore, every (n + 2)-fold Pfister form is isotropic, so I^{n+2}(F) = 0, i.e., st(F) ≤ n + 1. If the characteristic of F is 0, then the cohomological 2-dimension of F (\sqrt{−1}) is at most n + 2 by [124, II.4.1, Prop. 10 and II.4.2, Prop. 11]. By Fact 16.2 and the Hauptsatz 23.7, we have I^{n+3}(F (\sqrt{−1})) = 0. Thus st(F) ≤ n+2. □

As many problems in a field F reduce to finitely many elements over its prime field, we can often reduce to a problem over a given field to another over a field having finite stable range. We then can try to solve the problem when the stable range is finite. We shall use this idea repeatedly below.

Exercise 41.2. Let K/F be a finite simple extension of degree r. If I^{n}(F) = 0, then I^{n+r}(K) = 0. In particular, if a field has finite stable range, then any finite extension also has finite stable range.

Next we study graded annihilators.

Let 𝔟 be a bilinear n-fold Pfister form. For any m ≥ 0 set

ann_{m}(𝔟) = {𝔞 ∈ I^{m}(F) | 𝔞 · 𝔟 = 0 ∈ W (F)},
\overline{ann}_{m}(𝔟) = \{𝔞̄ ∈ \overline{I}^{m}(F) | 𝔞̄ · 𝔟̄ = 0 ∈ GW (F)\}.

Similarly, for a quadratic n-fold Pfister form ρ and any m ≥ 0 set

ann_{m}(ρ) = {𝔞 ∈ I^{m}(F) | 𝔞 · ρ = 0 ∈ I_{q}(F)},
\overline{ann}_{m}(ρ) = \{𝔞̄ ∈ \overline{I}^{m}(F) | 𝔞̄ · ρ̄ = 0 ∈ \overline{I}_{q}(F)\}.

It follows from Corollary 6.23 and Theorem 9.12 that ann_{1}(𝔟) and ann_{1}(ρ) are generated by the binary forms in them. Thus the following theorem determines completely the graded annihilators.

Theorem 41.3. Let 𝔟 and ρ be bilinear and quadratic n-fold Pfister forms respectively. Then for any m ≥ 1, we have

ann_{m}(𝔟) = I^{m−1}(F) · ann_{1}(𝔟), \overline{ann}_{m}(𝔟) = \overline{I}^{m−1}(F) · \overline{ann}_{1}(𝔟),
ann_{m}(ρ) = I^{m−1}(F) · ann_{1}(ρ), \overline{ann}_{m}(ρ) = \overline{I}^{m−1}(F) · \overline{ann}_{1}(ρ).

Proof. The case char F = 2 is proven in [14, Th.1.1 and 1.2]. We assume that char F ≠ 2. It is sufficient to consider the case of the bilinear form 𝔟.

It follows from Fact 40.7 and Fact 16.2 that the sequence

∐ \overline{I}^{m}(L) \underrightarrow{∑ s_{∗}} \overline{I}^{m}(F) ⟶^{·𝔟} \overline{I}^{n+m}(F),

is exact where the direct sum is taken over all quadratic field extensions L/F with 𝔟_{L} isotropic. By Lemma 34.16, we have I^{m}(L) = I^{m−1}(F)I(L), hence the image of s_{∗} : I^{m}(L) → I^{m}(F) is contained in I^{m−1}(F) · ann_{1}(𝔟). Therefore, the kernel of the second map in the sequence coincides with the image of I^{m−1}(F) · ann_{1}(𝔟) in \overline{I}^{m}(F). This proves \overline{ann}_{m}(𝔟) = \overline{I}^{m−1}(F) · \overline{ann}_{1}(𝔟).

Let 𝔠 ∈ ann_{m}(𝔟). We need to show that 𝔠 ∈ I^{m−1}(F) · ann_{1}(𝔟). We may assume that F is finitely generated over its prime field and hence F has finite stable range by Lemma 41.1. Let k be an integer such that k + m > st(F). Repeatedly applying exactness of the sequence above, we see that 𝔠 is congruent to an element 𝔞 ∈ I^{k+m}(F) modulo I^{m−1}(F) · ann_{1}(𝔟). Replacing 𝔠 by 𝔞 we may assume that m > st(F).

We claim that it suffices to prove the result for 𝔠 an m-fold Pfister form. By Theorem 33.15, for any 𝔠 ∈ I^{m}(F), there is an integer n such that

2^{n} sgn 𝔠 = ∑_{i=1}^{r} k_{i} · sgn 𝔠_{i},

with k_{i} ∈ ℤ and (n + m)-fold Pfister forms 𝔠_{i} with pairwise disjoint supports. As m > st(F), it follows from Proposition 35.22 that 𝔠_{i} ≃ 2^{n}𝔡_{i} for some m-fold Pfister forms 𝔡_{i}. Since I^{m}(F) is torsion-free, we have

𝔠 = ∑_{i=1}^{r} k_{i} · 𝔡_{i}

in I^{m}(F) and the supports of the 𝔡_{i}’s are pairwise disjoint. In particular, if 𝔟 ⊗ 𝔠 is hyperbolic, then supp(𝔟) ∩ supp(𝔠) = ∅, so supp(𝔟) ∩ supp(𝔡_{i}) = ∅ for every i. As I^{m}(F) is torsion-free, this would mean that 𝔟 ⊗ 𝔠_{i} is hyperbolic for every i and establishes the claim. Therefore, we may assume that 𝔠 is a Pfister form.

The result now follows from Lemma 35.18(1). □

We turn to the generators for I_{t}^{n}(F), the torsion in I^{n}(F).

Theorem 41.4. For any field F we have I_{t}^{n}(F) = I^{n−1}(F)I_{t}(F).

Proof. Let 𝔠 ∈ I_{t}^{n}(F). Then 2^{m}𝔠 = 0 for some m. Applying Theorem 41.3 to the Pfister form 𝔟 = 2^{m}⟨1⟩, we have

𝔠 ∈ ann_{n}(𝔟) = I^{n−1}(F) · ann_{1}(𝔟) ⊂ I^{n−1}(F)I_{t}(F).  □

Recall that by Proposition 31.27, the group I_{t}(F) is generated by binary torsion forms. Hence Theorem 41.4 yields

Corollary 41.5. A field F satisfies property A_{n} if and only if I^{n}(F) is torsion-free.

Remark 41.6. By Theorem 41.4, every torsion bilinear n-fold Pfister form 𝔟 can be written as a ℤ-linear combination of the (torsion) forms ⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩ with a_{1} ∈ D(∞⟨1⟩). Note that 𝔟 itself may not be isometric to a form like this (cf. Example 32.4).

Theorem 41.7. Let 𝔟 and ρ be bilinear and quadratic n-fold Pfister forms respectively. Then for any m ≥ 0, we have

W (F)𝔟 ∩ I^{n+m}(F) = I^{m}(F)𝔟,
W (F)ρ ∩ I_{q}^{n+m}(F) = I^{m}(F)ρ.

Proof. We prove the first equality (the second being similar). Let 𝔠 ∈ W (F)𝔟∩ I^{n+m}(F). We show by induction on m that 𝔠 ∈ I^{m}(F)𝔟. Suppose that 𝔠 = 𝔞 · 𝔟 in W (F) for some 𝔞 ∈ I^{m−1}(F), i.e., 𝔞̄ ∈ \overline{ann}_{m−1}(𝔟). By Theorem 41.3, we have 𝔞̄ = 𝔡̄𝔢̄ for some 𝔡 ∈ I^{m−2}(F) and 𝔢 ∈ W (F) satisfying 𝔢̄ ∈ \overline{ann}_{1}(𝔟). Let 𝔣 be a binary bilinear form congruent to 𝔢 modulo I^{2}(F). As 𝔣̄𝔟̄ = 𝔢̄𝔟̄ = 0 ∈ \overline{I}^{n+1}(F), the general (n + 1)-fold Pfister form 𝔣 ⊗ 𝔟 belongs to I^{n+2}(F). By the Hauptsatz 23.7, we have 𝔣 · 𝔟 = 0 in W (F). Since 𝔞 ≡ 𝔡𝔣 modulo I^{m}(F) it follows that 𝔠 = 𝔞𝔟 ∈ I^{m}(F)𝔟. □

Exercise 41.8. Let 𝔟 and 𝔠 be bilinear k-fold and n-fold Pfister forms, respectively, over a field F of characteristic not 2. Prove that for any m ≥ 1 the group

W (F)𝔠 ∩ ann_{W (F)}(𝔟) ∩ I^{m+n}(F)

is generated by (m + n)-fold Pfister forms 𝔡 in ann_{W (F)}(𝔟) that are divisible by 𝔠.

The theorem allows us to answer the problems at the end of §33. The solution to Lam’s problem in [88] was proven in [8] and independently by Dickmann and Miraglia (cf. [29]).

Corollary 41.9. Let 𝔟 be a form over F. If 2^{n}𝔟 ∈ I^{n+m}(F), then 𝔟 ∈ I^{m}(F) + W_{t}(F). In particular,

sgn(𝔟) ∈ C(𝔛(F), 2^{m}ℤ) if and only if 𝔟 ∈ I^{m}(F) + W_{t}(F).

Proof. Suppose that sgn 𝔟 ∈ C(𝔛(F), 2^{m}ℤ). By Theorem 33.15, there exists a form 𝔞 ∈ I^{n+m}(F) such that 2^{n} sgn 𝔟 = sgn 𝔞 for some n. In particular, 2^{n}𝔟 − 𝔞 ∈ W_{t}(F). Therefore, 2^{k+n}𝔟 = 2^{k}𝔞 for some k. By Theorem 41.7, applied to the form 2^{k+n}⟨1⟩, we may write 2^{k}𝔞 = 2^{k+n}𝔠 for some 𝔠 ∈ I^{n}(F). Then 𝔟 − 𝔠 lies in W_{t}(F) as needed. □

As a consequence, we obtain a solution to Marshall’s question in [97]. It was first proved by Dickmann and Miraglia in [28].

Corollary 41.10. Let F be a formally real pythagorean field. Let 𝔟 be a form over F. If 2^{n}𝔟 ∈ I^{n+m}(F), then 𝔟 ∈ I^{m}(F). In particular, sgn(I^{m}(F)) = C(𝔛(F), 2^{m}ℤ).

If F is a formally real field, let GC(𝔛(F), ℤ) be the graded ring

GC(𝔛(F), ℤ) := ∐ 2^{n}C(𝔛(F), ℤ)/2^{n+1}C(𝔛(F), ℤ) = ∐ C(𝔛(F), 2^{n}ℤ/2^{n+1}ℤ)

and GW_{t}(F) the graded ideal in GW (F) induced by I_{t}(F). Then Corollary 41.9 implies that the signature induces an exact sequence

0 → GW_{t}(F) → GW (F) → GC(𝔛(F), ℤ)

and Corollary 41.10 says if F is a formally real pythagorean field, then the signature induces an isomorphism GW (F) → GC(𝔛(F), ℤ).

We interpret this result in terms of the reduced Witt ring and prove the result mentioned at the end of §34.

Theorem 41.11. Let K be a quadratic extension and s : K → F a nonzero F - linear functional such that s(1) = 0. Then the sequence

0 → I_{red}^{n}(K/F) → I_{red}^{n}(F) ⟶^{r_{K/F}} I_{red}^{n}(K) ⟶^{s_{∗}} I_{red}^{n}(F)

is exact.

Proof. We need only to show exactness at I_{red}^{n}(K). Let 𝔠 ∈ I_{red}^{n}(K) satisfy s_{∗}(𝔠) is trivial in I_{red}^{n}(F), i.e., the form s_{∗}(𝔠) is torsion. By Theorem 41.4, we have s_{∗}(𝔠) = ∑ 𝔞_{i}𝔟_{i} with 𝔞_{i} ∈ I^{n−1}(F) and 𝔟_{i} ∈ I_{t}(F). It follows by Corollary 34.32 that 𝔟_{i} = s_{∗}(𝔡_{i}) for some torsion forms 𝔡_{i} ∈ I(K). Therefore, the form 𝔢 := 𝔠−∑(𝔞_{i})_{K}𝔡_{i} belongs to the kernel of s_{∗} : I^{n}(K) → I^{n}(F). It follows from Theorems 40.3 and 40.5 that 𝔢 = r_{K/F} (𝔣) for some 𝔣 ∈ I^{n}(F). Therefore, 𝔠 ≡ r_{K/F} (𝔣) modulo torsion. □

42. Presentation of I^{n}(F)

In this section, using the validity of the Milnor Conjecture, we show that the presentation established for I^{2}(F) in Theorem 4.22 generalizes to a presentation for I^{n}(F). This was first proven in [8] in the case of characteristic not 2. The characteristic 2 case was also proven independently by Arason and Baeza from that given below in [6].

Let n ≥ 2 and let \underline{I}_{n}(F) be the abelian group generated by all the isometry classes [𝔟] of bilinear n-fold Pfister forms 𝔟 subject to the generating relations:

(1) [⟨⟨1, 1, …, 1⟩⟩] = 0.

(2) [⟨⟨ab, c⟩⟩⊗𝔡]+[⟨⟨a, b⟩⟩⊗𝔡] = [⟨⟨a, bc⟩⟩⊗𝔡]+[⟨⟨b, c⟩⟩⊗𝔡] for all a, b, c ∈ F^{×} and bilinear (n − 2)-fold Pfister forms 𝔡.

Note that the group \underline{I}_{2}(F) was defined earlier in Section §4.C.

There is a natural surjective group homomorphism g_{n} : \underline{I}_{n}(F) → I^{n}(F) taking the class [𝔟] of a bilinear n-fold Pfister form 𝔟 to 𝔟 in I^{n}(F). The map g_{2} is an isomorphism by Theorem 4.22.

As in the proof of Lemma 4.18, applying both relations repeatedly, we find that [⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩] = 0 if a_{1} = 1. It follows that for any bilinear m-fold Pfister form 𝔟, the assignment 𝔞 ↦ 𝔞 ⊗ 𝔟 gives rise to a well-defined homomorphism

\underline{I}_{n}(F) → \underline{I}_{n+m}(F)

taking [𝔞] to [𝔞 ⊗ 𝔟].

Lemma 42.1. Let 𝔟 be a metabolic bilinear n-fold Pfister form. Then [𝔟] = 0 in \underline{I}_{n}(F).

42. PRESENTATION OF I^{n}(F) 185

Proof. We prove the statement by induction on n. Since g_{2} is an isomorphism, the statement is true if n = 2. In the general case, we write 𝔟 = ⟨⟨a⟩⟩ ⊗ 𝔠 for some a ∈ F^{×} and bilinear (n − 1)-fold Pfister form 𝔠. We may assume by induction that 𝔠 is anisotropic. It follows from Corollary 6.14 that 𝔠 ≃ ⟨⟨b⟩⟩ ⊗ 𝔡 for some b ∈ F^{×} and bilinear (n − 2)-fold Pfister form 𝔡 such that ⟨⟨a, b⟩⟩ is metabolic. By the case n = 2, we have [⟨⟨a, b⟩⟩] = 0 in \underline{I}_{2}(F), hence [𝔟] = [⟨⟨a, b⟩⟩ ⊗ 𝔡] = 0 in \underline{I}_{n}(F). □

For each n, let α_{n} : \underline{I}_{n+1}(F) → \underline{I}_{n}(F) be the map given by
[⟨⟨a, b⟩⟩ ⊗ 𝔠] ↦ [⟨⟨a⟩⟩ ⊗ 𝔠] + [⟨⟨b⟩⟩ ⊗ 𝔠] − [⟨⟨ab⟩⟩ ⊗ 𝔠].

We show that this map is well-defined. Let ⟨⟨a, b⟩⟩ ⊗ 𝔠 and ⟨⟨a′, b′⟩⟩ ⊗ 𝔠′ be isometric bilinear n-fold Pfister forms. We need to show that

(42.2) [⟨⟨a⟩⟩ ⊗ 𝔠] + [⟨⟨b⟩⟩ ⊗ 𝔠] − [⟨⟨ab⟩⟩ ⊗ 𝔠] = [⟨⟨a′⟩⟩ ⊗ 𝔠′] + [⟨⟨b′⟩⟩ ⊗ 𝔠′] − [⟨⟨a′b′⟩⟩ ⊗ 𝔠′]

in \underline{I}_{n}(F). By Theorem 6.10, the forms ⟨⟨a, b⟩⟩ ⊗ 𝔠 and ⟨⟨a′, b′⟩⟩ ⊗ 𝔠′ are chain pequivalent. Thus we may assume that one of the following cases hold:

(1) a = a′, b = b′ and 𝔠 ≃ 𝔠′.

(2) ⟨⟨a, b⟩⟩ ≃ ⟨⟨a′, b′⟩⟩ and 𝔠 = 𝔠′.

(3) a = a′, 𝔠 = ⟨⟨c⟩⟩ ⊗ 𝔡, and 𝔠′ = ⟨⟨c′⟩⟩ ⊗ 𝔡 for some c ∈ F^{×} and bilinear

(n − 2)-fold Pfister form 𝔡 and ⟨⟨b, c⟩⟩ ≃ ⟨⟨b′, c′⟩⟩.

It follows that it is sufficient to prove the statement in the case n = 2. The equality (42.2) holds if we compose the morphism α_{2} with the homomorphism g_{2} : \underline{I}_{2}(F) → I^{2}(F). But g_{2} is an isomorphism, hence α_{n} is well-defined. It is easy to check that α_{n} is a homomorphism.

The homomorphism α_{n} fits in the commutative diagram

\underline{I}_{n+1}(F) ⟶^{α_{n}} \underline{I}_{n}(F)
g_{n+1}↓  ↓^{g_{n}}
I^{n+1}(F) ⟶ I^{n}(F)

with the bottom map the inclusion.

Lemma 42.3. The natural homomorphism

γ : Coker(α_{n}) → \overline{I}^{n}(F)

is an isomorphism.

Proof. Consider the map

τ : (F^{×})^{n} → Coker(α_{n})

given by (a_{1}, a_{2}, …, a_{n}) ↦ [⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩] + im(α_{n}). Clearly τ is symmetric with respect to permutations of the a_{i}’s.

By definition of α_{n} we have

[⟨⟨a⟩⟩ ⊗ 𝔠] + [⟨⟨b⟩⟩ ⊗ 𝔠] ≡ [⟨⟨ab⟩⟩ ⊗ 𝔠] mod im(α_{n})

for any bilinear (n − 1)-fold Pfister form 𝔠. It follows that τ is multilinear.

The map τ also satisfies the Steinberg relation. Indeed, if a_{1} + a_{2} = 1, then [⟨⟨a_{1}, a_{2}⟩⟩] = 0 in \underline{I}_{2}(F) as g_{2} is an isomorphism and therefore [⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩] = 0 in \underline{I}_{n}(F).

As the group Coker(α_{n}) has exponent 2, the map τ induces a group homomorphism

k_{n}(F) = K_{n}(F)/2K_{n}(F) → Coker(α_{n})

which we also denote by τ. The composition γ ∘ τ takes a symbol {a_{1}, a_{2}, …, a_{n}} to ⟨⟨a_{1}, a_{2}, …, a_{n}⟩⟩ + I^{n+1}(F). By Fact 5.15, the map γ ∘ τ is an isomorphism. As τ is surjective, we have γ is an isomorphism. □

It follows from Lemma 42.3 that we have a commutative diagram

\underline{I}_{n+1}(F) ⟶^{α_{n}} \underline{I}_{n}(F) ⟶ \overline{I}^{n}(F) ⟶ 0
g_{n+1}↓  ^{g_{n}}↓  ∥
0 ⟶ I^{n+1}(F) ⟶ I^{n}(F) ⟶ \overline{I}^{n}(F) ⟶ 0

with exact rows. It follows that if g_{n+1} is an isomorphism, then g_{n} is also an isomorphism.

Theorem 42.4. If n ≥ 2, the abelian group I^{n}(F) is generated by the isometry classes of bilinear n-fold Pfister forms subject to the generating relations

(1) ⟨⟨1, 1, …, 1⟩⟩ = 0.

(2) ⟨⟨ab, c⟩⟩ · 𝔡 + ⟨⟨a, b⟩⟩ · 𝔡 = ⟨⟨a, bc⟩⟩ · 𝔡 + ⟨⟨b, c⟩⟩ · 𝔡 for all a, b, c ∈ F^{×} and bilinear (n − 2)-fold Pfister forms 𝔡.

Proof. We shall show that the surjective map g_{n} : \underline{I}_{n}(F) → I^{n}(F) is an isomorphism. Any element in the kernel of g_{n} = g_{n,F} belongs to the image of the natural map g_{n,F′} → g_{n,F} where F′ is a subfield of F, finitely generated over the prime subfield of F. Thus we may assume that F is finitely generated. It follows from Lemma 41.1 that F has finite stable range. The discussion preceding the theorem shows that we may also assume that n > st(F).

If F is not formally real, then I^{n}(F) = 0, i.e., every bilinear n-fold Pfister form is metabolic. By Lemma 42.1, the group \underline{I}_{n}(F) is trivial and we are done.

In what follows we may assume that F is formally real, in particular, char F ≠ 2. We let M be the abelian group given by generators {𝔟}, the isometry classes

of bilinear n-fold Pfister forms 𝔟 over F, and relations {𝔟} = {𝔠} + {𝔡} where the bilinear n-fold Pfister forms 𝔟, 𝔠 and 𝔡 satisfy 𝔟 = 𝔠 + 𝔡 in W (F). In particular, {𝔟} = 0 in M if 𝔟 = 0 in W (F).

We claim that the homomorphism

δ : M → \underline{I}_{n}(F) given by \{𝔟\} ↦ [𝔟]

is well-defined. To see this, it suffices to check that if 𝔟, 𝔠 and 𝔡 satisfy 𝔟 = 𝔠 + 𝔡 in W (F), then [𝔟] = [𝔠] + [𝔡] in \underline{I}_{n}(F). As char F ≠ 2, it follows from Proposition 24.5 that there are c, d ∈ F^{×} and a bilinear (n − 1)-fold Pfister form 𝔞 such that

𝔠 ≃ ⟨⟨c⟩⟩ ⊗ 𝔞, 𝔡 ≃ ⟨⟨d⟩⟩ ⊗ 𝔞, 𝔟 ≃ ⟨⟨cd⟩⟩ ⊗ 𝔞.

The equality 𝔟 = 𝔠 + 𝔡 implies that ⟨⟨c, d⟩⟩ · 𝔞 = 0 in W (F). Therefore,

0 = α_{n}([⟨⟨c, d⟩⟩ ⊗ 𝔞]) = [𝔠] + [𝔡] − [𝔟]

in \underline{I}_{n}(F), hence the claim.

Let 𝔟 be a bilinear n-fold Pfister form and d ∈ F^{×}. As I^{n+1}(F) = 2I^{n}(F), we can write ⟨⟨d⟩⟩ · 𝔟 = 2𝔠 and ⟨⟨−d⟩⟩ · 𝔟 = 2𝔡 in W (F) with 𝔠, 𝔡 bilinear n-fold Pfister forms. Adding, we then get 2𝔟 = 2𝔠 + 2𝔡 in W (F), hence 𝔟 = 𝔠 + 𝔡 since I^{n}(F) is torsion-free. It follows that [𝔟] = [𝔠] + [𝔡] in M. We generalize this as follows:

Lemma 42.5. Let F be a formally real field having finite stable range. Suppose that n is a positive integer ≥ st(F). Let 𝔟 ∈ P_{n}(F) and d_{1}, …, d_{m} ∈ F^{×}. For every ε = (ε_{1}, …, ε_{m}) ∈ {±1}^{m} write ⟨⟨ε_{1}d_{1}, …, ε_{m}d_{m}⟩⟩ ⊗ 𝔟 ≃ 2^{m}𝔠_{ε} with 𝔠_{ε} ∈ P_{n}(F). Then [𝔟] = ∑_{ε}[𝔠_{ε}] in M.

Proof. We induct on m: The case m = 1 was done above. So we assume that m > 1. For every ε′ = (ε_{2}, …, ε_{m}) ∈ {±1}^{m−1} write

⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩ ⊗ 𝔟 ≃ 2^{m−1}𝔡_{ε′}

with 𝔡_{ε′} ∈ P_{n}(F). By the induction hypothesis, we then have [𝔟] = ∑_{ε′}[𝔡_{ε′}] in M. It therefore suffices to show that [𝔡_{ε′}] = [𝔠_{(1,ε′)}] + [𝔠_{(−1,ε′)}] for every ε′. But

2^{m}𝔡_{ε′} = 2⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩ · 𝔟
= (⟨⟨d_{1}⟩⟩ + ⟨⟨−d_{1}⟩⟩) · ⟨⟨ε_{2}d_{2}, …, ε_{m}d_{m}⟩⟩ · 𝔟
= 2^{m}𝔠_{(1,ε′)} + 2^{m}𝔠_{(−1,ε′)}

in W (F), hence 𝔡_{ε′} = 𝔠_{(1,ε′)} + 𝔠_{(−1,ε′)} in W (F). Consequently, [𝔡_{ε′}] = [𝔠_{(1,ε′)}] + [𝔠_{(−1,ε′)}] in M. □

Proposition 42.6. Let F be a formally real field having finite stable range. Suppose that n is a positive integer ≥ st(F). Then every element in M can be written as a ℤ-linear combination ∑_{j=1}^{s} l_{j} · [𝔠_{j}] with forms 𝔠_{1}, …, 𝔠_{s} ∈ P_{n}(F) having pairwise disjoint supports in 𝔛(F).

Proof. Let 𝔞 = ∑_{i=1}^{r} k_{i} · [𝔟_{i}] ∈ M. Write 𝔟_{i} ≃ ⟨⟨a_{i1}, …, a_{in}⟩⟩ for i = 1, …, r.

For every matrix ε = (ε_{ik})_{i=1,k=1}^{r,n} in {±1}^{r×n} let 𝔣_{ε} ≃ ⨂_{j=1}^{r} ⨂_{l=1}^{n}⟨⟨ε_{jl}a_{jl}⟩⟩ and

write 𝔣_{ε} ⊗ 𝔟_{i} ≃ 2^{rn}𝔠_{i,ε} with 𝔠_{i,ε} bilinear n-fold Pfister forms for i ∈ [1, r]. By Lemma 42.5, we have [𝔟_{i}] = ∑_{ε}[𝔠_{i,ε}] in M for i ∈ [1, r], hence

𝔞 = ∑_{i=1}^{r} k_{i} · [𝔟_{i}] = ∑_{i=1}^{r} k_{i} · ∑_{ε}[𝔠_{i,ε}] = ∑_{ε} ∑_{i=1}^{r} k_{i} · [𝔠_{i,ε}]

in M.

For each ε, write 𝔣_{ε} ≃ 2^{nr−n}𝔡_{ε} with 𝔡_{ε} a bilinear n-fold Pfister form. Clearly, the 𝔣_{ε} have pairwise disjoint supports, hence also the 𝔡_{ε}. Now look at a pair (i, ε). If all the ε_{ik}, k ∈ [1, r], are 1, then 𝔣_{ε} ⊗ 𝔟_{i} = 2^{n}𝔣_{ε} = 2^{nr}𝔡_{ε}, hence 𝔠_{i,ε} = 𝔡_{ε}. If, however, some ε_{ik}, k ∈ [1, r], is −1, then 𝔣_{ε} ⊗ 𝔟_{i} = 0, hence 𝔠_{i,ε} = 0. It follows that for each ε we have ∑_{i=1}^{r} k_{i} · [𝔠_{i,ε}] = l_{ε} · 𝔡_{ε} for some integer l_{ε}. Consequently,

𝔞 = ∑_{ε} ∑_{i=1}^{r} k_{i} · [𝔠_{i,ε}] = ∑_{ε} l_{ε} · 𝔡_{ε}.  □

Applying Proposition 42.6 to an element in the kernel of the composition

M ⟶^{δ} \underline{I}_{n}(F) ⟶^{g_{n}} I^{n}(F) ⟶^{sgn} C(𝔛(F), ℤ),

we see that all the coefficients l_{j} are 0. Hence the composition is injective. Since δ is surjective, it follows that g_{n} is injective and therefore an isomorphism. The proof of Theorem 42.4 is complete. □

Remark 42.7. In [6], Arason and Baeza also establish a presentation of I_{q}^{n}(F). They show that as an additive group I_{q}^{n}(F) for n ≥ 2 is isomorphic to the equivalence classes {φ} of n-fold quadratic Pfister forms φ over F subject to the relations:

(1_{q}) {α⊗[1, d_{1} +d_{2}]} = {α⊗[1, d_{1}]}+{α⊗[1, d_{2}]} for any (n−1)-fold bilinear Pfister form α and d_{1}, d_{2} ∈ F.

(2_{q}) {⟨⟨a⟩⟩ ⊗ ρ} + {⟨⟨b⟩⟩ ⊗ ρ} = {⟨⟨a + b⟩⟩ ⊗ ρ} + {⟨⟨ab(a + b)⟩⟩ ⊗ ρ} for any (n − 1)-fold quadratic Pfister form ρ and a, b ∈ F^{×} satisfying a + b ≠ 0.

43. Going down and torsion-freeness

We show in this section that if K/F is a finite extension with I^{n}(K) torsionfree, then I^{n}(F) is torsion-free. Since we already know this to be true if char F = 2 by Lemma 35.5, we need only show this when char F ≠ 2. In this case, we show that the solution of the Milnor Conjecture that the norm residue map is an isomorphism implies the result as shown in [7] and, in fact, leads to the more general Corollary 43.10 below.

Let F be a field of characteristic not 2. For any integer k, n ≥ 0 consider Galois cohomology groups (cf. §101)

H^{n}(F, k) := H^{n,n−1}(F, ℤ/2^{k}ℤ).

In particular, H^{n}(F, 1) = H^{n}(F).

By Corollary 101.7, there is an exact sequence

0 → H^{n}(F, r) → H^{n}(F, r + s) → H^{n}(F, s).

For a field extension L/F set

H^{n}(L/F, k) := Ker(H^{n}(F, k) ⟶^{r_{L/F}} H^{n}(L, k)).

For all r, s ≥ 0, we have an exact sequence

(43.1)  0 → H^{n}(L/F, r) → H^{n}(L/F, r + s) → H^{n}(L/F, s).

Proposition 43.2. Let char F ≠ 2. Suppose I_{t}^{n}(F) = 0. Then H^{n}(F_{py}/F, k) = 0 for all k.

Proof. Let α ∈ H^{n}(F_{py}/F). As F_{py} is the union of admissible extensions over F (cf. Definition 31.13), there is an admissible sub-extension L/F of F_{py}/F such that α_{L} = 0. We induct on the degree [L : F] to show α_{L} = 0. Let E be a subfield of L such that E/F is admissible and L = E(\sqrt{d}) with d ∈ D(2⟨1⟩_{E}). It follows from the exactness of the cohomology sequence (cf. Theorem 99.13) for the quadratic extension L/E that α_{E} ∈ H^{n−1}(E) ∪ (d). By Proposition 35.7, the field E satisfies A_{n}. Hence all the torsion Pfister forms ⟨⟨a_{1}, …, a_{n−1}, d⟩⟩ over E are trivial, consequently, H^{n−1}(E) ∪ (d) = 0 by Fact 16.2 and therefore α_{E} = 0. By the induction hypothesis, α = 0.

We have shown that H^{n}(F_{py}/F) = 0. Triviality of the group H^{n}(F_{py}/F, k) follows then by induction on k from exactness of the sequence (43.1). □

Exercise 43.3. Let char F ≠ 2. Show that if H^{n}(F_{py}/F) = 0, then I^{n}(F) is torsion-free.

Lemma 43.4. A field F of characteristic different from 2 is pythagorean if and only if F has no cyclic extensions of degree 4.

Proof. Consider the exact sequence

H^{1}(F, 2) ⟶^{g} H^{1}(F) ⟶^{b} H^{2}(F),

where b is the Bockstein homomorphism, b((a)) = (a) ∪ (−1) (cf. Proposition 101.14). The field F is not pythagorean if and only if there is nonsquare a ∈ F^{×} such that a ∈ D(2⟨1⟩). The latter is equivalent to (a)∪(−1) = 0 in H^{2}(F) = Br_{2}(F) which in its turn is equivalent to (a) ∈ im(g), i.e., the quadratic extension F (\sqrt{a})/F can be embedded into a cyclic extension of degree 4. □

Let F be a field of characteristic different from 2 such that μ_{2^{n}} ⊂ F with n > 1 and m ≤ n. Then Kummer theory implies that the natural map

(43.5)  F^{×}/F^{×2^{n}} = H^{1}(F, n) → H^{1}(F, m) = F^{×}/F^{×2^{m}}

is surjective.

Lemma 43.6. Let F be a pythagorean field of characteristic different from 2. Then

c_{F (\sqrt{−1})/F} : H^{1}(F (\sqrt{−1}), s) → H^{1}(F, s)

is trivial for every s.

Proof. If F is nonreal, then it is quadratically closed, so H^{1}(F, s) = 0. Therefore, we may assume that F is formally real. In particular, F (\sqrt{−1}) ≠ F.

Let β ∈ H^{1}(F, s + 1) = Hom_{cont}(Γ_{F} , ℤ/2^{s+1}ℤ). Then the kernel of β is an open subgroup U of Γ_{F} with Γ_{F} /U cyclic of 2-power order. As F is pythagorean, F has no cyclic extensions of a 2-power order greater than 2 by Lemma 43.4. It follows that [Γ_{F} : U] ≤ 2, hence β lies in the image of H^{1}(F) → H^{1}(F, s + 1). Consequently, β lies in the kernel of H^{1}(F, s + 1) → H^{1}(F, s). This shows that the natural map H^{1}(F, s + 1) → H^{1}(F, s) is trivial. The statement now follows from the commutativity of the diagram

H^{1}(F (\sqrt{−1}), s + 1) ⟶^{c_{F (\sqrt{−1})/F)}} H^{1}(F, s + 1)
↓  ↓^{0}
H^{1}(F (\sqrt{−1}), s) ⟶^{c_{F (\sqrt{−1})/F)}} H^{1}(F, s)

together with the surjectivity of H^{1}(F (\sqrt{−1}), s+1) → H^{1}(F (\sqrt{−1}), s) which holds by (43.5) as μ_{2^{∞}} ⊂ ℚ_{py}(\sqrt{−1}) ⊂ F (\sqrt{−1}). □

Lemma 43.7. Let F be a field of characteristic different from 2 satisfying μ_{2^{s}} ⊂ F (\sqrt{−1}). Then for every d ∈ D(2⟨1⟩) the class (d) belongs to the image of the natural map H^{1}(F_{py}/F, s) → H^{1}(F_{py}/F).

Proof. By (43.5), the natural map g : H^{1}(F (\sqrt{−1}), s) → H^{1}(F (\sqrt{−1})) is surjective. As d ∈ N_{F (\sqrt{−1})/F} (F (\sqrt{−1})), there exists a γ ∈ H^{1}(F (\sqrt{−1}), s) satisfying (d) = g(c_{F (\sqrt{−1})/F} (γ)). By Lemma 43.6, we have c_{F (\sqrt{−1})/F} (γ) ∈ H^{1}(F_{py}/F, s) and the image of c_{F (\sqrt{−1})/F} (γ) in H^{1}(F_{py}/F) coincides with (d). □

Theorem 43.8. Let char F ≠ 2 and K/F a finite field extension. If I^{n}(K) is torsion-free for some n, then I^{n}(F) is also torsion-free.

Proof. Let 2^{r} be the largest power of 2 dividing [K : F]. Suppose first that the field F (\sqrt{−1}) contains μ_{2^{r+1}}.

By Corollary 41.5, we know that I^{n}(F) is torsion-free if and only if F satisfies A_{n}. By Lemma 35.2, it suffices to show any bilinear n-fold Pfister form ⟨⟨a_{1}, …, a_{n−1}, d⟩⟩ with a_{1}, …, a_{n−1} ∈ F^{×} and d ∈ D(2⟨1⟩) is hyperbolic. By Lemma 43.7, there is an α ∈ H^{1}(F_{py}/F, r + 1) such that the natural map

H^{1}(F_{py}/F, r + 1) → H^{1}(F_{py}/F)

takes α to (d).

Recall that the graded group H^{∗}(F_{py}/F, r + 1) has the natural structure of a module over the Milnor ring K_{∗}(F) (cf. 101.5). Consider the element

β = {a_{1}, …, a_{n−1}} · α ∈ H^{n}(F_{py}/F, r + 1).

As I_{t}^{n}(K) = 0, we have H^{n}(K_{py}/K, r + 1) = 0 by Proposition 43.2. Therefore,

[K : F] · β = c_{K/F} ∘ r_{K/F} (β) = 0,

hence 2^{r}β = 0. The composition

H^{n}(F, r + 1) → H^{n}(F) → H^{n}(F, r + 1)

coincides with the multiplication by 2^{r}. Since the second homomorphism is injective by (43.1), the image {a_{1}, …, a_{n−1}}·(d) = (a_{1}, …, a_{n−1}, d) of β in H^{n}(F) is trivial. Therefore, ⟨⟨a_{1}, …, a_{n−1}, d⟩⟩ is hyperbolic by Fact 16.2.

Consider the general case. As μ_{2^{∞}} ⊂ F_{py}(\sqrt{−1}) there is a subfield E ⊂ F_{py} such that μ_{2^{r+1}} ⊂ E(\sqrt{−1}) and E/F is an admissible extension. Then L := KE is an admissible extension of K. In particular, I^{n}(L) is torsion-free by Proposition 35.7 and Corollary 41.5. Note also that the degree [L : E] divides [K : F]. By the first part of the proof applied to the extension L/E we have I_{t}^{n}(E) = 0. It follows from Theorem 35.12 and Corollary 41.5 that I_{t}^{n}(F) = 0. □

Corollary 43.9. Let K be a finite extension of a nonformally real field F. If I^{n}(K) = 0, then I^{n}(F) = 0.

Proof. If char F = 2, this was shown in Lemma 35.5. If char F ≠ 2, this follows from Theorem 43.8. □

Define the torsion-free index of a field F to be

ν(F) := min {n | I^{n}(F) is torsion-free}

(or infinity if no such integer exists). Then we have

Corollary 43.10. Let K/F a finitely generated field extension. Then

ν(F) + tr. deg(K/F) ≤ ν(K).

Proof. If K/F is finite, then ν(F) ≤ ν(K) by Theorem 43.8. It follows from this and induction that we may assume that K = F (t). If 𝔟 ∈ I_{t}^{n}(F) where ν(K) ≤ n + 1, then ⟨⟨t⟩⟩ · 𝔟 = 0 in W (K) as it is torsion in I^{n+1}(K). It follows by Example 19.13 that 𝔟 = 0 in W (F). □

Remark 43.11. Let F be a field of characteristic not 2.

(1) Define the absolute stability index st_{a}(F) to be the minimum n (if it exists) such that I^{n+1}(F) = 2I^{n}(F). Corollary 35.27 implies that

ν(F (\sqrt{−1})) ≤ n + 1 if and only if ν(F) ≤ n + 1 and st_{a}(F) ≤ n.

(which, by Remark 35.28, is equivalent to I^{n+1}(L) = 0 for some quadratic extension L/F.) By Fact 16.2 this is equivalent to H^{n+1}(F (\sqrt{−1})) = 0.

(2) Let K/F be a finitely generated field extension of transcendental degree m. In [41] it is shown that if m > 0, then H^{n+1}(F (\sqrt{−1})) = 0 is equivalent to cd_{2}(K(\sqrt{−1})) ≤ n + 1. This is not true for K/F algebraic, e.g., let F be the quadratic closure of ℚ and K = F (\sqrt[3]{2}).

(3) The “going up” problem of the relationship of ν(F) and ν(K) even when K/F is finite is unclear. In [37] it was shown that ν(K) ≤ ν(F) + [K : F] − 1 when F is not formally real and that ν(K) ≥ 1 + ν(F) is possible, but no uniform bound is known even when F is quadratically closed. Leep (unpublished) has improved this bound to ν(K) ≤ ν(F) + [log_{2}([K : F]/3)] + 1 (for F ≠ K). Similarly, it can be shown that st_{a}(K) ≤ st_{a}(F) + [K : F] − 1, but again no uniform bound is known. This compares to the relative case where Bröcker showed in [19] using valuation theory that the reduced stability satisfied st_{r}(F) + m ≤ st_{r}K ≤ st_{r}(F) + tr. deg(K/F) + 1.

CHAPTER VIII On the Norm Residue Homomorphism of Degree Two

44. The main theorem

In this chapter we prove the degree two case of the Milnor Conjecture (cf. Fact 101.6).

Theorem 44.1. For every field F of characteristic not 2, the norm residue homomorphism

h_{F} := h_{F}^{2} : K_{2}(F)/2K_{2}(F) → Br_{2}(F),

taking \{a, b\} + 2K_{2}(F) to the class of the quaternion algebra \binom{a,b}{F} is an isomorphism.

Corollary 44.2. Let F be a field of characteristic not 2. Then

(1) The group Br_{2}(F) is generated by the classes of quaternion algebras.

(2) The following is the list of the defining relations between classes of quaternion algebras:

(i) \binom{aa′,b}{F} = \binom{a,b}{F} · \binom{a′,b}{F}; \binom{a,bb′}{F} = \binom{a,b}{F} · \binom{a,b′}{F},

(ii) \binom{a,b}{F} · \binom{a,b}{F} = 1,

(iii) \binom{a,b}{F} = 1 if a + b = 1.

The main idea of the proof is to compare the norm residue homomorphisms h_{F} and h_{F (C)}, where C is a smooth conic curve over F. The function field F (C) is a generic splitting field for a symbol in k_{2}(F) := K_{2}(F)/2K_{2}(F), so passing from F to F (C) allows us to carry out inductive arguments.

Theorem 44.1 was originally proven in [100]. The proof used a specialization argument reducing the problem to the study of the function field of a conic curve and a comparison theorem of Suslin [128] on the behavior of the norm residue homomorphism over the function field of a conic curve.

The “elementary” proof presented in this chapter relies neither on a specialization argument nor on higher K-theory. The key point of the proof is Theorem 46.1. It is also a consequence of Quillen’s computation of higher K-theory of a conic curve [113, §8, Th. 4.1] and a theorem of Rehmann and Stuhler on the group K_{2} of a quaternion algebra given in [114].

Other “elementary” proofs of the bijectivity of h_{F}, avoiding higher K-theory, but still using a specialization argument, were given by Arason in [5] and Wadsworth in [138].

45. Geometry of conic curves

In this section we establish interrelations between projective conic curves and their corresponding quaternion algebras over a field F of arbitrary characteristic.

45.A. Quaternion algebras and conic curves. Let Q be a quaternion algebra over a field F. Recall (cf. Appendix 98.E) that Q carries a canonical involution a ↦ ā, the reduced trace linear map

Trd : Q → F, a ↦ a + ā

and the reduced norm quadratic map

Nrd : Q → F, a ↦ aā.

Every element a ∈ Q satisfies the quadratic equation

a^{2} − Trd(a)a + Nrd(a) = 0.

Set

V_{Q} := Ker(Trd) = {a ∈ Q | ā = −a},

so V_{Q} is a 3-dimensional subspace of Q. Note that x^{2} = − Nrd(x) ∈ F for any x ∈ V_{Q}, and the map φ_{Q} : V_{Q} → F given by φ_{Q}(x) = x^{2} is a quadratic form on V_{Q}. The space V_{Q} is the orthogonal complement to 1 in Q with respect to the nondegenerate bilinear form on Q:

(a, b) ↦ Trd(ab).

The quadric C_{Q} of the form φ_{Q}(x) in the projective plane ℙ(V_{Q}) is a smooth projective conic curve. In fact, every smooth projective conic curve (1-dimensional quadric) is of the form C_{Q} for some quaternion algebra Q (cf. Exercise 12.6).

Proposition 45.1. The following conditions are equivalent:

(1) Q is split.

(2) C_{Q} is isomorphic to the projective line ℙ^{1}.

(3) C_{Q} has a rational point.

Proof. (1) ⇒ (2): The algebra Q is isomorphic to the matrix algebra 𝕄_{2}(F). Hence V_{Q} is the space of trace 0 matrices and C_{Q} is given by the equation t_{0}^{2}+t_{1}t_{2} = 0. The morphism C_{Q} → ℙ^{1}, given by [t_{0} : t_{1} : t_{2}] ↦ [t_{0} : t_{1}] = [−t_{2} : t_{0}] is an isomorphism.

(2) ⇒ (3) is obvious.

(3) ⇒ (1): There is a nonzero element x ∈ Q such that x^{2} = 0. In particular, Q is not a division algebra and therefore Q is split. □

Corollary 45.2. Let Q be a quaternion algebra. Then (1) Every divisor on C_{Q} of degree zero is principal.

(2) If Q is a division algebra, the degree of every closed point of C_{Q} is even.

Proof. (1): Let w be a divisor on C_{Q} of degree zero. Let K/F be a separable quadratic splitting field of Q and σ the nontrivial automorphism of K over F. By Proposition 45.1, the conic C_{Q} is isomorphic to the projective line over K. As deg σ(w_{K}) = deg w_{K}, there is a function f ∈ K(C_{Q})^{×} with div(f) = w_{K}. Let s = f · σ(f^{−1}). Since div(s) = w_{K} − σ(w_{K}) = 0 we have s ∈ K^{×} and N_{K/F} (s) = s · σ(s) = 1. By the Hilbert Theorem 90, there is t ∈ K^{×} with s = σ(t) · t^{−1}. Setting g = tf we have σ(g) = g, i.e., g ∈ F (C_{Q})^{×} and div(g) = w.

(2): Let x ∈ C_{Q} be a closed point. As C_{Q} has a rational point over F (x), by Proposition 45.1, the algebra Q is split over F (x). It follows from Corollary 98.5 that the degree [F (x) : F] is even. □

Example 45.3. If char F ≠ 2, there is a basis 1, i, j, k of Q such that a := i^{2} ∈ F^{×}, b := j^{2} ∈ F^{×}, k = ij = −ji (cf. Example 98.10). Then V_{Q} = F i ⊕ F j ⊕ F k and C_{Q} is given by the equation at_{0}^{2} + bt_{1}^{2} − abt_{2}^{2} = 0.

Example 45.4. If char F = 2, there is a basis 1, i, j, k of Q such that a := i^{2} ∈ F, b := j^{2} ∈ F, k = ij = ji + 1 (cf. Example 98.11). Then V_{Q} = F 1 ⊕ F i ⊕ F j and C_{Q} is given by the equation t_{0}^{2} + at_{1}^{2} + bt_{2}^{2} + t_{1}t_{2} = 0.

For every a ∈ Q define the F -linear function l_{a} on V_{Q} by the formula

l_{a}(x) = Trd(ax).

The reduced trace form Trd is a nondegenerate bilinear form on Q. Indeed, to see this it is sufficient to check over a splitting field of Q where Q is isomorphic to a matrix algebra, and it is clear in this case. Hence every F -linear function on V_{Q} is equal to l_{a} for some a ∈ Q.

Lemma 45.5. Let a, b ∈ Q and α, β ∈ F. Then:

(1) l_{a} = l_{b} if and only if a − b ∈ F.

(2) l_{αa+βb} = αl_{a} + βl_{b}.

(3) l\overline{a} = −l_{a}.

(4) l_{a^{−1}} = −(Nrd a)^{−1} · l_{a} if a is invertible.

Proof. (1): This follows from the fact that V_{Q} is orthogonal to F with respect to the bilinear form Trd.

(2) is obvious.

(3): For any x ∈ V_{Q}, we have

l\overline{a}(x) = Trd(āx) = Trd(\overline{āx}) = Trd(x̄a) = − Trd(xa) = − Trd(ax) = −l_{a}(x).

(4): It follows from (2) and (3) that (Nrd a)l_{a^{−1}} = l\overline{a} = −l_{a}. □

Every element a ∈ Q \ F generates a quadratic subalgebra F [a] = F ⊕ F a of Q. Conversely, every quadratic subalgebra K of Q is of the form F [a] for any a ∈ K \F. By Lemma 45.5, the linear form l_{a} on V_{Q} is independent, up to a multiple in F^{×}, of the choice of a ∈ K \ F. Hence the line in ℙ(V_{Q}) given by the equation l_{a}(x) = 0 is determined by K. The intersection of this line with the conic C_{Q} is an effective divisor on C_{Q} of degree two. Thus, we have the following maps:

Quadratic ; subalgebras ; of Q → Rational ; points ; in ℙ(V_{Q}^{∗}) = \begin{matrix}Lines \\ in ℙ(V_{Q}) \end{matrix} → Degree 2 ; effective ; divisors ; on C_{Q}

Proposition 45.6. The two maps above are bijections.

Proof. The first map is a bijection since every line in ℙ(V_{Q}) is given by an equation l_{a} = 0 for some a ∈ Q \ F and a generates a quadratic subalgebra of Q. The last map is a bijection since the embedding of C_{Q} as a closed subscheme of ℙ(V_{Q}) is given by a complete linear system. □

Remark 45.7. Effective divisors on C_{Q} of degree 2 are rational points of the symmetric square S^{2}(C_{Q}). Proposition 45.6 essentially asserts that S^{2}(C_{Q}) is isomorphic to the projective plane ℙ(V_{Q}^{∗}).

Suppose Q is a division algebra. Then the conic curve C_{Q} has no rational points and quadratic subalgebras of Q are quadratic (maximal) subfields of Q. An effective cycle on C_{Q} of degree 2 is a closed point of degree 2. Thus, by Proposition 45.6, we have bijections

Quadratic ; subfields of Q →^{∼} Rational points in ℙ(V_{Q}^{∗}) = \begin{matrix}Lines \\ in ℙ(V_{Q}) \end{matrix} →^{∼} Points of degree 2 in C_{Q}

In what follows, we shall frequently use the bijection constructed above between the set of quadratic subfields of Q and the set of degree 2 closed points of C_{Q}.

45.B. Key identity. In the following proposition, we write a multiple of the quadratic form φ_{Q} on V_{Q} as a degree two polynomial of linear forms.

Proposition 45.8. Let Q be a quaternion algebra over F. For any a, b, c ∈ Q,

l_{a\overline{b}} · l_{c} + l_{b\overline{c}} · l_{a} + l_{c\overline{a}} · l_{b} = (Trd(cba) − Trd(abc)) · φ_{Q}.

Proof. We write T for Trd in this proof. For every x ∈ V_{Q} we have:

l_{a\overline{b}}(x) · l_{c}(x) = T (a\overline{b}x)T (cx)

= T (a(T (b) − b)x)T (cx)

= T (ax)T (b)T (cx) − T (abx)T (cx)

= T (ax)T (b)T (cx) − T (abT (cx)x)

= T (ax)T (b)T (cx) − T (abc)x^{2} + T (abx\overline{c}x),

l_{b\overline{c}}(x) · l_{a}(x) = T (b\overline{c}x)T (ax)

= T ((T (b) − \overline{b})\overline{c}x)T (ax)

= T (\overline{c}x)T (b)T (ax) − T (\overline{b}\overline{c}x)T (ax)

= −T (ax)T (b)T (cx) − T (\overline{b}\overline{c}x(ax + x̄ā))

= −T (ax)T (b)T (cx) − T (\overline{b}\overline{c}xax) + T (\overline{b}\overline{c}\overline{a})x^{2}

= −T (ax)T (b)T (cx) − T (ax\overline{b}\overline{c}x) + T (cba)x^{2}

l_{c\overline{a}}(x) · l_{b}(x) = T (c\overline{a}x)T (bx)

= −T (a\overline{c}x)T (bx)

= −T (aT (bx)\overline{c}x)

= −T (abx\overline{c}x) + T (ax\overline{b}\overline{c}x).

Adding the equalities yields the result. □

45.C. Residue fields of points of C_{Q} and quadratic subfields of Q. Suppose that the quaternion algebra Q is a division algebra. Recall that quadratic subfields of Q correspond bijectively to degree 2 points of C_{Q}. We shall show how to identify a quadratic subfield of Q with the residue field of the corresponding point in C_{Q} of degree 2.

Choose a quadratic subfield K ⊂ Q. For every a ∈ Q\K, one has Q = K ⊕aK. We define a map

μ_{a} : V_{Q}^{∗} → K

by the rule: if c = u + av for u, v ∈ K, then μ_{a}(l_{c}) = v. Clearly,

μ_{a}(l_{c}) = 0  if and only if  c ∈ K.

By Lemma 45.5, the map μ_{a} is well-defined and F -linear. If b ∈ Q \ K is another element, we have

(45.9)  μ_{b}(l_{c}) = μ_{b}(l_{a})μ_{a}(l_{c}),

hence the maps μ_{a} and μ_{b} differ by the multiple μ_{b}(l_{a}) ∈ K^{×}. The map μ_{a} extends to an F -algebra homomorphism

μ_{a} : S^{•}(V_{Q}^{∗}) → K

in the usual way (where S^{•} denotes the symmetric algebra).

Let x ∈ C_{Q} ⊂ ℙ(V_{Q}) be the point of degree 2 corresponding to the quadratic subfield K. The local ring O_{ℙ(V_{Q}),x} is the subring of the quotient field of the symmetric algebra S^{•}(V_{Q}^{∗}) generated by the fractions l_{c}/l_{d} for all c ∈ Q and d ∈ Q \ K.

Fix an element a ∈ Q \ F. We define an F -algebra homomorphism

μ : O_{ℙ(V_{Q}),x} → K

by the formula

μ(l_{c}/l_{d}) = μ_{a}(l_{c})/μ_{a}(l_{d}).

Note that μ_{a}(l_{d}) ≠ 0 since d ∉ K and the map μ is independent of the choice of a ∈ Q \ K by (45.9).

We claim that the map μ vanishes on the quadratic form φ_{Q} defining C_{Q} in ℙ(V_{Q}). Proposition 45.8 gives a formula for a multiple of the quadratic form φ_{Q} with the coefficient α := Trd(cba) − Trd(abc).

Lemma 45.10. There exist a ∈ Q \ K, b ∈ K, and c ∈ Q such that α ≠ 0.

Proof. Pick any b ∈ K \ F and any a ∈ Q such that ab ≠ ba. Clearly, a ∈ Q \ K. Then α = Trd((ba − ab)c) is nonzero for some c ∈ Q since the bilinear form Trd is nondegenerate on Q. □

Choose a, b and c as in Lemma 45.10. We have μ_{a}(l_{b}) = 0 since b ∈ K. Also, μ_{a}(l_{a}) = 1 and μ_{a}(l_{a\overline{b}}) = b̄. Write c = u + av for u, v ∈ K, then μ_{a}(l_{c}) = v. As

bc̄ = bū + bv̄ā = bū + Trd(bv̄ā) − avb̄,

we have μ_{a}(l_{b\overline{c}}) = −vb̄ and by Proposition 45.8,

αμ(φ_{Q}) = μ_{a}(l_{a\overline{b}})μ_{a}(l_{c}) + μ_{a}(l_{b\overline{c}})μ_{a}(l_{a}) + μ_{a}(l_{c\overline{a}})μ_{a}(l_{b}) = b̄v − vb̄ = 0.

Since Q is a division algebra and α ≠ 0, we have μ(φ_{Q}) = 0 as claimed. The local ring O_{C_{Q},x} coincides with the factor ring

O_{ℙ(V_{Q}),x}/φ_{Q}O_{ℙ(V_{Q}),x}.

Therefore, μ factors through an F -algebra homomorphism

μ : O_{C_{Q},x} → K.

Let e ∈ K \ F. The function l_{e}/l_{a} is a local parameter of the local ring O_{C_{Q},x}, i.e., it generates the maximal ideal of O_{C_{Q},x}. Since μ(l_{e}/l_{a}) = 0, the map μ induces a field isomorphism

(45.11)  F (x) →^{∼} K

of degree 2 field extensions of F. We have proved

Proposition 45.12. Let Q be a division quaternion algebra. Let K ⊂ Q be a quadratic subfield and x ∈ C_{Q} the corresponding point of degree 2. Then the residue field F (x) is canonically isomorphic to K over F. Let a ∈ Q and b ∈ Q \ K. Write a = u + bv for unique u, v ∈ K. Then the value (l_{a}/l_{b})(x) in F (x) of the function l_{a}/l_{b} at the point x corresponds to the element v ∈ K under the isomorphism (45.11).

46. Key exact sequence

In this section we prove exactness of a sequence comparing the groups K_{2}(F) and K_{2}(F (C)) over a field F of arbitrary characteristic.

Let C be a smooth curve over a field F. For every (closed) point x ∈ C, there is a residue homomorphism

∂_{x} : K_{2}(F (C)) → K_{1}(F (x)) = F (x)^{×}

induced by the discrete valuation of the local ring O_{C,x} (cf. §49.A).

In this section we prove the following:

Theorem 46.1. Let C be a conic curve over a field F. Then the sequence

K_{2}(F) → K_{2}(F (C)) ⟶^{∂} ∐_{x∈C} F (x)^{×} ⟶^{c} F^{×},

with ∂ = (∂_{x}) and c = (N_{F (x)/F}), is exact.

46.A. Filtration on K_{2}(F (C)). Let C be a conic over F. If C has a rational point, i.e., C ≃ ℙ_{F}^{1} , the statement of Theorem 46.1 is Milnor’s computation of K_{2}(F (t)) given in Theorem 100.7. So we may (and will) assume that C has no rational point. By Corollary 45.2, the degree of every closed point of C is even.

Fix a closed point x_{0} ∈ C of degree 2. As in §29, for any n ∈ ℤ let L_{n} be the F -subspace

{f ∈ F (C)^{×} | div(f) + nx_{0} ≥ 0} ∪ {0}

of F (C). Clearly, L_{n} = 0 if n < 0. Recall that L_{0} = F and L_{n} · L_{m} ⊂ L_{n+m}. It follows from Lemma 29.7 that dim L_{n} = 2n + 1 if n ≥ 0.

We write L_{n}^{×} for L_{n} \ {0}. Note that the value g(x) in F (x) is defined for every g ∈ L_{n}^{×} and point x ≠ x_{0}.

Since any divisor on C of degree zero is principal by Corollary 45.2, for every point x ∈ C of degree 2n we can choose a function p_{x} ∈ L_{n}^{×} satisfying div(p_{x}) = x − nx_{0}. In particular, p_{x_{0}} ∈ F^{×}. Note that p_{x} is uniquely determined up to a scalar multiple. Clearly, p_{x}(x) = 0 if x ≠ x_{0}. Every function in L_{n}^{×} can be written as the product of a nonzero constant and finitely many p_{x} for some points x of degree at most 2n.

Lemma 46.2. Let x ∈ C be a point of degree 2n different from x_{0}. If g ∈ L_{m} satisfies g(x) = 0, then g = p_{x}q for some q ∈ L_{m−n}. In particular, g = 0 if m < n.

Proof. Consider the F -linear map

e_{x} : L_{m} → F (x), e_{x}(g) = g(x).

If m < n, the map e_{x} is injective since x does not belong to the support of the divisor of a function in L_{m}^{×}. Suppose that m = n and g ∈ Ker e_{x}. Then div(g) = x − nx_{0} and hence g is a multiple of p_{x}. Thus, the kernel of e_{x} is 1-dimensional. By dimension count, e_{x} is surjective.

Therefore, for arbitrary m ≥ n, the map e_{x} is surjective and

dim Ker(e_{x}) = dim L_{m} − deg(x) = 2m + 1 − 2n.

The image of the injective linear map L_{m−n} → L_{m} given by multiplication by p_{x} is contained in Ker(e_{x}) and of dimension dim L_{m−n} = 2m + 1 − 2n. Therefore, Ker(e_{x}) = p_{x}L_{m−n}. □

For every n ∈ ℤ, let M_{n} be the subgroup of K_{2}(F (C)) generated by the symbols {f, g} with f, g ∈ L_{n}^{×}, i.e., M_{n} = {L_{n}^{×}, L_{n}^{×}}. We have the following filtration:

(46.3)  0 = M_{−1} ⊂ M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ K_{2}(F (C)).

Note that M_{0} coincides with the image of the homomorphism K_{2}(F) → K_{2}(F (C)) and K_{2}(F (C)) is the union of all the M_{n} as the group F (C)^{×} is the union of the subsets L_{n}^{×}.

If f ∈ L_{n}^{×}, the degree of every point in the support of div(f) is at most 2n. In particular, ∂_{x}(M_{n−1}) = 0 for every point x of degree 2n. Therefore, for every n ≥ 0, we have a well-defined homomorphism

∂_{n} : M_{n}/M_{n−1} → ∐_{deg x=2n} F (x)^{×}

induced by ∂_{x} over all points x ∈ C of degree 2n.

We refine the filtration (46.3) by adding an extra term M′ between M_{0} and M_{1}. Set M′ := {L_{1}^{×}, L_{0}^{×}} = {L_{1}^{×}, F^{×}}, so the group M′ is generated by M_{0} and symbols of the form {p_{x}, α} for all points x ∈ C of degree 2 and all α ∈ F^{×}.

Denote by A′ the subgroup of ∐_{deg x=2} F (x)^{×} consisting of all families (α_{x}) such that α_{x} ∈ F^{×} for all x and ∏_{x} α_{x} = 1. Clearly, ∂_{1}(M′/M_{0}) ⊂ A′.

Theorem 46.1 is a consequence of the following three propositions.

Proposition 46.4. If n ≥ 2, the map

∂_{n} : M_{n}/M_{n−1} → ∐_{deg x=2n} F (x)^{×}

is an isomorphism.

Proposition 46.5. The restriction ∂′ : M′/M_{0} → A′ of ∂_{1} is an isomorphism.

Proposition 46.6. The sequence

0 → M_{1}/M′ ⟶^{∂_{1}} ( ∐_{deg x=2} F (x)^{×}) /A′ ⟶^{c} F^{×}

is exact.

Proof of Theorem 46.1. Since K_{2}(F (C)) is the union of M_{n}, it is sufficient to prove that the sequence

0 → M_{n}/M_{0} ⟶^{∂} ∐_{deg x≤2n} F (x)^{×} ⟶^{c} F^{×}

is exact for every n ≥ 1. We induct on n. The case n = 1 follows from Propositions 46.5 and 46.6. The induction step is guaranteed by Proposition 46.4. □

46.B. Proof of Proposition 46.4. To effect the proof, we shall construct the inverse map of ∂_{n}. We need the following two lemmas.

Lemma 46.7. Let x ∈ C be a point of degree 2n > 2. Then for every u ∈ F (x)^{×}, there exist f ∈ L_{n−1}^{×} and h ∈ L_{1}^{×} such that (f/h)(x) = u.

Proof. The F -linear map e_{x} : L_{n−1} → F (x), e_{x}(f) = f(x), is injective by Lemma 46.2. Hence

dim Coker(e_{x}) = deg(x) − dim L_{n−1} = 2n − (2n − 1) = 1.

Consider the F -linear map

g : L_{1} → Coker(e_{x}), g(h) = u · h(x) + Im(e_{x}).

Since dim L_{1} = 3, the kernel of g contains a nonzero function h ∈ L_{1}^{×}. We have u · h(x) = f(x) for some f ∈ L_{n−1}^{×}. Since deg x > 2 the value h(x) is nonzero. Hence u = (f/h)(x). □

Let x ∈ C be a point of degree 2n > 2. Define a map

ψ_{x} : F (x)^{×} → M_{n}/M_{n−1}

as follows: By Lemma 46.7, for each element u ∈ F (x)^{×}, we can choose f ∈ L_{n−1}^{×} and h ∈ L_{1}^{×} such that (f/h)(x) = u. We set

ψ_{x}(u) = \{p_{x}, \frac{f}{h} \} + M_{n−1}.

Lemma 46.8. The map ψ_{x} is a well-defined homomorphism.

Proof. Let f′ ∈ L_{n−1}^{×} and h′ ∈ L_{1}^{×} be two functions with (f′/h′)(x) = u. Then f′h − fh′ ∈ L_{n} and (f′h − fh′)(x) = 0. By Lemma 46.2, we have f′h − fh′ = λp_{x} for some λ ∈ F. If λ = 0, then f/h = f′/h′.

Suppose λ ≠ 0. Since (λp_{x})/(f′h) + (fh′)/(f′h) = 1, we have

0 = \{\frac{λp_{x}}{f′h} , \frac{fh′}{f′h}\} ≡ \{p_{x}, \frac{f}{h} \} − \{p_{x}, \frac{f′}{h′} \} mod M_{n−1}.

Hence

\{p_{x}, \frac{f}{h} \} + M_{n−1} = \{p_{x}, \frac{f′}{h′} \} + M_{n−1},

so that the map ψ is well-defined.

Let u_{3} = u_{1}u_{2} ∈ F (x)^{×}. Choose f_{i} ∈ L_{n−1}^{×} and h_{i} ∈ L_{1}^{×} satisfying (f_{i}/h_{i})(x) = u_{i} for i = 1, 2, 3. The function f_{1}f_{2}h_{3} −f_{3}h_{1}h_{2} belongs to L_{2n−1} and has zero value at x. We have f_{1}f_{2}h_{3} − f_{3}h_{1}h_{2} = p_{x}q for some q ∈ L_{n−1} by Lemma 46.2. Since (p_{x}q)/(f_{1}f_{2}h_{3}) + (f_{3}h_{1}h_{2})/(f_{1}f_{2}h_{3}) = 1,

0 = \{\frac{p_{x}q}{f_{1}f_{2}h_{3}} , \frac{f_{3}h_{1}h_{2}}{f_{1}f_{2}h_{3}} \} ≡ \{p_{x}, \frac{f_{3}}{h_{3}} \} − \{p_{x}, \frac{f_{1}}{h_{1}} \} − \{p_{x}, \frac{f_{2}}{h_{2}} \} mod M_{n−1}.

Thus, ψ_{x}(u_{3}) = ψ_{x}(u_{1}) + ψ_{x}(u_{2}). □

By Lemma 46.8, we have a homomorphism

ψ_{n} = ∑ ψ_{x} : ∐_{deg x=2n} F (x)^{×} → M_{n}/M_{n−1}.

We claim that ∂_{n} and ψ_{n} are isomorphisms inverse to each other. If x is a point of degree 2n > 2 and u ∈ F (x)^{×}, choose f ∈ L_{n−1}^{×} and h ∈ L_{1}^{×} such that (f/h)(x) = u. We have

∂_{x}(\{p_{x}, \frac{f}{h} \}) = (\frac{f}{h} )(x) = u

and the symbol {p_{x}, f/h} has no nontrivial residues at other points of degree 2n. Therefore, ∂_{n} ∘ ψ_{n} is the identity.

To finish the proof of Proposition 46.4, it suffices to show that ψ_{n} is surjective. The group M_{n}/M_{n−1} is generated by classes of the form {p_{x}, g} + M_{n−1} and {p_{x}, p_{y}} + M_{n−1}, where g ∈ L_{n−1}^{×} and x, y are distinct points of degree 2n. Clearly,

{p_{x}, g} + M_{n−1} = ψ_{x}(g(x)),

hence {p_{x}, g} + M_{n−1} ∈ Im ψ_{n}.

By Lemma 46.7, there are elements f ∈ L_{n−1}^{×} and h ∈ L_{1}^{×} such that p_{x}(y) = (f/h)(y). The function p_{x}h−f belongs to L_{n+1}^{×} and has zero value at y. Therefore, p_{x}h − f = p_{y}q for some q ∈ L_{1}^{×} by Lemma 46.2. Since (p_{y}q)/(p_{x}h) + f/(p_{x}h) = 1 we have

0 = \{\frac{p_{y}q}{p_{x}h}, \frac{f}{p_{x}h}\} ≡ \{p_{x}, p_{y}\} mod Im(ψ_{n}). □

46.C. Proof of Proposition 46.5. Define a homomorphism

ρ : A′ → M′/M_{0}

by the rule

ρ( ∐ α_{x}) = ∑_{deg x=2}{p_{x}, α_{x}} + M_{0}.

Since ∂_{x}{p_{x}, α} = α and ∂_{x_{0}}{p_{x}, α} = α^{−1} for every x ≠ x_{0} and the product of all α_{x} is equal to 1, the composition ∂′ ∘ ρ is the identity. Clearly, ρ is surjective. □

46.D. Generators and relations of A(Q)/A′. It remains to prove Proposition 46.6. Let Q be a quaternion division algebra satisfying C →^{∼} C_{Q}. By Proposition 45.12, the norm homomorphism

∐_{deg x=2} F (x)^{×} → F^{×}

is canonically isomorphic to the norm homomorphism

(46.9)  ∐ K^{×} → F^{×},

where the coproduct is taken over all quadratic subfields K ⊂ Q. Note that the norm map N_{K/F} : K^{×} → F^{×} is the restriction of the reduced norm Nrd on K. Let A(Q) be the kernel of the norm homomorphism (46.9). Under the above canonical isomorphism the subgroup A′ of ∐ F (x)^{×} corresponds to the subgroup of A(Q) (that we still denote by A′) consisting of all families (a_{K}) satisfying a_{K} ∈ F^{×} and ∏ a_{K} = 1, i.e., A′ is the intersection of A(Q) and ∐ F^{×}. Therefore, Proposition 46.6 asserts that the canonical homomorphism

(46.10)  ∂_{1} : M_{1}/M′ → A(Q)/A′

is an isomorphism. In the proof of Proposition 46.6, we shall construct the inverse isomorphism. In order to do so, it is convenient to have a presentation of the group A(Q)/A′ by generators and relations.

We define a map (not a homomorphism!)

Q^{×} → ( ∐ K^{×})/A′, a ↦ ã

as follows: If a ∈ Q^{×} is not a scalar, it is contained in a unique quadratic subfield K of Q. Therefore, a defines an element of the coproduct ∐ K^{×}. We denote by ã the corresponding class in (∐ K^{×})/A′. If a ∈ F^{×}, of course, a belongs to all quadratic subfields. Nevertheless, a defines a unique element ã of the factor group (∐ K^{×})/A′ (we place a in any quadratic subfield). Clearly,

(46.11)  ˜(ab) = ã · b̃ if a and b commute.

(Note that we are using multiplicative notation for the operation in the factor group.) Obviously, the group (∐ K^{×})/A′ is an abelian group generated by the ã for all a ∈ Q^{×} with the set of defining relations given by (46.11).

The group A(Q)/A′ is generated (as an abelian group) by the products

ã_{1}ã_{2} ⋯ ã_{n} with a_{i} ∈ Q^{×} and Nrd(a_{1}a_{2} ⋯ a_{n}) = 1,

with the following set of defining relations:

(1) (ã_{1}ã_{2} ⋯ ã_{n}) · (ã_{n+1}ã_{n+2} ⋯ ã_{n+m}) = (ã_{1}ã_{2} ⋯ ã_{n+m}).

(2) ãb̃(ã^{−1})(b̃^{−1}) = 1.

(3) If a_{i−1} and a_{i} commute, then ã_{1} ⋯ ã_{i−1}ã_{i} ⋯ ã_{n} = ã_{1} ⋯ { \atop a_{i−1}a_{i}} ⋯ ã_{n}.

The set of generators is too large for our purposes. In the next subsection, we shall find another presentation of A(Q)/A′ (cf. Corollary 46.27). More precisely, we shall define an abstract group G by generators and relations (with a “better” set of generators) and prove that G is isomorphic to A(Q)/A′.

46.E. The group G. Let Q be a division quaternion algebra over a field F. Consider the abelian group G defined by generators and relations as follows: The sign ∗ will denote the operation in G with 1 its identity element. Generators: Symbols (a, b, c) for all ordered triples with a, b, c elements of Q^{×} satisfying abc = 1. Note that if (a, b, c) is a generator of G, then so are the cyclic permutations (b, c, a) and (c, a, b). Relations:

(R1): (a, b, cd)∗(ab, c, d) = (b, c, da)∗(bc, d, a) for all a, b, c, d ∈ Q^{×} satisfying abcd = 1;

(R2): (a, b, c) = 1 if a and b commute.

For an (ordered) sequence a_{1}, a_{2}, …, a_{n} (n ≥ 1) of elements in Q^{×} satisfying a_{1}a_{2} … a_{n} = 1, we define a symbol

(a_{1}, a_{2}, …, a_{n}) ∈ G

by induction on n as follows. The symbol is trivial if n = 1 or 2. If n ≥ 3, we set

(a_{1}, a_{2}, …, a_{n}) := (a_{1}, a_{2}, …, a_{n−2}, a_{n−1}a_{n}) ∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}).
Note that if a_{1}a_{2} … a_{n} = 1, then a_{2} … a_{n}a_{1} = 1.

Lemma 46.12. The symbols do not change under cyclic permutations, i.e., (a_{1}, a_{2}, …, a_{n}) = (a_{2}, …, a_{n}, a_{1}) if a_{1}a_{2} … a_{n} = 1.

Proof. We induct on n. The statement is clear if n = 1 or 2. If n = 3,

(a_{1}, a_{2}, a_{3}) = (a_{1}, a_{2}, a_{3}) ∗ (a_{1}a_{2}, a_{3}, 1) (relation R2)
= (a_{2}, a_{3}, a_{1}) ∗ (a_{2}a_{3}, 1, a_{1}) (relation R1)
= (a_{2}, a_{3}, a_{1}) (relation R2).

Suppose that n ≥ 4. We have

(a_{1}, a_{2}, …, a_{n}) = (a_{1}, …, a_{n−2}, a_{n−1}a_{n}) ∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}) (definition)
=(a_{2}, …, a_{n−2}, a_{n−1}a_{n}, a_{1}) ∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}) (induction)
=(a_{2}, …, a_{n−2}, a_{n−1}a_{n}a_{1}) ∗ (a_{2}a_{3} ⋯ a_{n−2}, a_{n−1}a_{n}, a_{1})
∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}) (definition)
=(a_{2}, …, a_{n−2}, a_{n−1}a_{n}a_{1}) ∗ (a_{1}, a_{2}a_{3} ⋯ a_{n−2}, a_{n−1}a_{n})
∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}) (case n = 3)
=(a_{2}, …, a_{n−2}, a_{n−1}a_{n}a_{1}) ∗ (a_{2}a_{3} ⋯ a_{n−2}, a_{n−1}, a_{n}a_{1})
∗ (a_{2}a_{3} ⋯ a_{n−1}, a_{n}, a_{1}) (relation R1)
=(a_{2}, …, a_{n−2}, a_{n−1}, a_{n}a_{1}) ∗ (a_{2}a_{3} ⋯ a_{n−1}, a_{n}, a_{1}) (definition)
=(a_{2}, …, a_{n}, a_{1}) (definition).  □

Lemma 46.13. If a_{1}a_{2} … a_{n} = 1 and a_{i−1} commutes with a_{i} for some i, then

(a_{1}, …, a_{i−1}, a_{i}, …, a_{n}) = (a_{1}, …, a_{i−1}a_{i}, …, a_{n}).

Proof. We may assume that n ≥ 3 and i = n by Lemma 46.12. We have

(a_{1}, …, a_{n−2}, a_{n−1}, a_{n})
= (a_{1}, …, a_{n−2}, a_{n−1}a_{n}) ∗ (a_{1}a_{2} ⋯ a_{n−2}, a_{n−1}, a_{n}) (definition)
= (a_{1}, …, a_{n−2}, a_{n−1}a_{n}) (relation R2).  □

Lemma 46.14. (a_{1}, …, a_{n}) ∗ (b_{1}, …, b_{m}) = (a_{1}, …, a_{n}, b_{1}, …, b_{m}).

Proof. We induct on m. By Lemma 46.13, we may assume that m ≥ 3. We have

L.H.S. = (a_{1}, …, a_{n}) ∗ (b_{1}, …, b_{m−1}b_{m}) ∗ (b_{1}b_{2} ⋯ b_{m−2}, b_{m−1}, b_{m}) (definition)
= (a_{1}, …, a_{n}, b_{1}, …, b_{m−1}b_{m}) ∗ (b_{1}b_{2} ⋯ b_{m−2}, b_{m−1}, b_{m}) (induction)
= (a_{1}, …, a_{n}, b_{1}, …, b_{m}) (definition).  □
As usual, we write [a, b] for the commutator aba^{−1}b^{−1}.

Lemma 46.15. Let a, b ∈ Q^{×}.

(1) For every nonzero b′ ∈ F b + F ba, one has [a, b] = [a, b′]. Similarly, [a, b] = [a′, b] for every nonzero a′ ∈ F a + F ab.

(2) For every nonzero b′ ∈ F b + F ba + F bab there exists a′ ∈ Q^{×} such that [a, b] = [a′, b] = [a′, b′].

Proof. (1): We have b′ = bx, where x ∈ F + F a. Hence x commutes with a, so [a, b] = [a, b′]. The proof of the second statement is similar.

(2): There is nonzero a′ ∈ F a + F ab such that b′ ∈ F b + F ba′. By the first part, [a, b] = [a′, b] = [a′, b′]. □

Corollary 46.16. (1) Let [a, b] = [c, d]. Then there are a′, b′ ∈ Q^{×} such that [a, b] = [a′, b] = [a′, b′] = [c, b′] = [c, d].

(2) Every pair of commutators in Q^{×} can be written in the form [a, b] and [c, d] with b = c.

Proof. (1): If [a, b] = 1 = [c, d], we can take a′ = b′ = 1. Otherwise, both sets {b, ba, bab} and {d, dc} are linearly independent. Let b′ be a nonzero element in the intersection of the subspaces F b + F ba + F bab and F d + F dc. The statement follows from Lemma 46.15.

(2): Let [a, b] and [c, d] be two commutators. We may clearly assume that [a, b] ≠ 1 ≠ [c, d], so that both sets {b, ba, bab} and {c, cd} are linearly independent. Choose a nonzero element b′ in the intersection of F b + F ba + F bab and F c + F cd. By Lemma 46.15, we have [a, b] = [a′, b′] for some a′ ∈ Q^{×} and [c, d] = [b′, d]. □

Lemma 46.17. Let h ∈ Q^{×}. The following conditions are equivalent:

(1) h = [a, b] for some a, b ∈ Q^{×}.

(2) h ∈ [Q^{×}, Q^{×}].

(3) Nrd(h) = 1.

Proof. The implications (1) ⇒ (2) ⇒ (3) are obvious.

(3) ⇒ (1): Let K be a separable quadratic subfield containing h. (If h is purely inseparable, then h^{2} ∈ F and therefore h = 1.) Since N_{K/F} (h) = Nrd(h) = 1, by the Hilbert Theorem 90, we have h = b̄b^{−1} for some b ∈ K^{×}. By the Noether-Skolem Theorem, b̄ = aba^{−1} for some a ∈ Q^{×}, hence h = [a, b]. □

Let h ∈ Q^{×} satisfy Nrd(h) = 1. Then h = [a, b] = aba^{−1}b^{−1} for some a, b ∈ Q^{×} by Lemma 46.17. Consider the following element

ĥ = (b, a, b^{−1}, a^{−1}, h) ∈ G.

Lemma 46.18. The element ĥ does not depend on the choice of a and b.

Proof. Let h = [a, b] = [c, d]. By Corollary 46.16(1), we may assume that either a = c or b = d. Consider the first case (the latter case is similar). We can write d = bx, where x commutes with a. We have

(d, a, d^{−1},a^{−1}, h) = (bx, a, x^{−1}b^{−1}, a^{−1}, h)
= (bx, x^{−1}, b^{−1}) ∗ (b, x, a, x^{−1}b^{−1}, a^{−1}, h) (Lemmas 46.13, 46.14)
= (bx, x^{−1}, b^{−1}) ∗ (a^{−1}, h, b, x, a, x^{−1}b^{−1}) (Lemma 46.12)
= (a^{−1}, h, b, x, a, x^{−1}b^{−1}, bx, x^{−1}, b^{−1}) (Lemma 46.14)
= (a^{−1}, h, b, a, b^{−1}) (Lemma 46.13)
= (b, a, b^{−1}, a^{−1}, h) (Lemma 46.12).  □

Lemma 46.19. For every h_{1}, h_{2} ∈ [Q^{×}, Q^{×}] we have

{ \atop h_{1}h_{2}} = ĥ_{1} ∗ ĥ_{2} ∗ (h_{1}h_{2}, h_{2}^{−1}, h_{1}^{−1}).

Proof. By Corollary 46.16(2), we have h_{1} = [a_{1}, c] and h_{2} = [c, b_{2}] for some a_{1}, b_{2}, c ∈ Q^{×}. Then h_{1}h_{2} = [a_{1}b_{2}^{−1}, b_{2}cb_{2}^{−1}] and

ĥ_{1} ∗ ĥ_{2} ∗ (h_{1}h_{2},h_{2}^{−1}, h_{1}^{−1})

= (c, a_{1}, c^{−1}, a_{1}^{−1}, h_{1}, h_{2}, b_{2}, c, b_{2}^{−1}, c^{−1}) ∗ (h_{1}h_{2}, h_{2}^{−1}, h_{1}^{−1})

= (b_{2}, c, b_{2}^{−1}, c^{−1}, c, a_{1}, c^{−1}, a_{1}^{−1}, h_{1}, h_{2}) ∗ (h_{2}^{−1}, h_{1}^{−1}h_{1}h_{2})

= (b_{2}, c, b_{2}^{−1}, a_{1}, c^{−1}, a_{1}^{−1}, h_{1}h_{2})

= (b_{2}, c, b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}) ∗ (b_{2}cb_{2}^{−1}, a_{1}, c^{−1}, a_{1}^{−1}, h_{1}h_{2})

= (b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}, b_{2}, c) ∗ (c^{−1}, a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1})

= (b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}, b_{2}, c, c^{−1}, a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1})

= (b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}, b_{2}, a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1}b_{2}^{−1}, b_{2}a_{1}^{−1}, a_{1})

= (b_{2}, a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1}b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}) ∗ (b_{2}a_{1}^{−1}, a_{1}, b_{2}^{−1})

= (b_{2}a_{1}^{−1}, a_{1}, b_{2}^{−1}, b_{2}, a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1}b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1})

= (b_{2}a_{1}^{−1}, h_{1}h_{2}, b_{2}cb_{2}^{−1}, a_{1}b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1})

= (b_{2}cb_{2}^{−1}, a_{1}b_{2}^{−1}, b_{2}c^{−1}b_{2}^{−1}, b_{2}a_{1}^{−1}, h_{1}h_{2})

= { \atop h_{1}h_{2}}.  □

Let a_{1}, a_{2}, …, a_{n} ∈ Q^{×} satisfying Nrd(h) = 1 where h = a_{1}a_{2} … a_{n}. We set

(46.20)  ((a_{1}, a_{2}, …, a_{n})) := (a_{1}, a_{2}, …, a_{n}, h^{−1}) ∗ ĥ ∈ G.

Lemma 46.21. ((a_{1}, a_{2}, …, a_{n})) ∗ ((b_{1}, b_{2}, …, b_{m})) = ((a_{1}, …, a_{n}, b_{1}, …, b_{m})).

Proof. Set h := a_{1} ⋯ a_{n} and h′ := b_{1} ⋯ b_{m}. We have

L.H.S. = (a_{1}, a_{2}, …, a_{n}, h^{−1}) ∗ (b_{1}, b_{2}, …, b_{m}, (h′)^{−1}) ∗ ĥ ∗ ĥ′

= (a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}, …, b_{m}, (h′)^{−1}, h^{−1}) ∗ ĥ ∗ ĥ′

= (a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}, …, b_{m}, (hh′)^{−1}) ∗ (hh′, (h′)^{−1}, h) ∗ ĥ ∗ ĥ′

= (a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}, …, b_{m}, (hh′)^{−1}) ∗ ĥh′ (Lemma 46.19)

= R.H.S.  □

The following lemma is a consequence of the definition (46.20) and Lemma 46.13.

Lemma 46.22. If a_{i−1} commutes with a_{i} for some i, then

((a_{1}, …, a_{i−1}, a_{i}, …, a_{n})) = ((a_{1}, …, a_{i−1}a_{i}, …, a_{n})).

Lemma 46.23. ((a, b, a^{−1}, b^{−1})) = 1.

Proof. Set h = [a, b]. We have

L.H.S. = (a, b, a^{−1}, b^{−1}, h^{−1}) ∗ ĥ = (a, b, a^{−1}, b^{−1}, h^{−1}) ∗ (b, a, b^{−1}, a^{−1}, h) = 1. □

We want to establish an isomorphism between G and A(Q)/A′. To do so we define a map π : G → A(Q)/A′ by the formula

π((a, b, c)) = ãb̃c̃ ∈ A(Q)/A′,

where a, b, c ∈ Q^{×} satisfy abc = 1. Clearly, π is well-defined.

Let a_{1}, a_{2}, …, a_{n} ∈ Q^{×} with a_{1}a_{2} ⋯ a_{n} = 1. By induction on n, we have
π((a_{1}, a_{2}, …, a_{n})) = ã_{1}ã_{2} ⋯ ã_{n} ∈ A(Q)/A′.

Hence π is a homomorphism by Lemma 46.14.

Let h ∈ [Q^{×}, Q^{×}]. Write h = [a, b] for a, b ∈ Q^{×}. We have

π(ĥ) = π((b, a, b^{−1}, a^{−1}, h)) = h̃.
If a_{1}, a_{2}, …, a_{n} ∈ Q^{×} satisfies Nrd(h) = 1 with h = a_{1}a_{2} ⋯ a_{n}, then
(46.24) π(((a_{1}, a_{2}, …, a_{n}))) = π((a_{1}, a_{2}, …, a_{n}, h^{−1})) ∗ π(ĥ) = ã_{1}ã_{2} ⋯ ã_{n}.

Define a homomorphism θ : A(Q)/A′ → G as follows: Let a_{1}, a_{2}, …, a_{n} ∈ Q^{×} satisfy Nrd(a_{1}a_{2} ⋯ a_{n}) = 1. Define θ by

(46.25)  θ(ã_{1}ã_{2} ⋯ ã_{n}) = ((a_{1}, a_{2}, …, a_{n})).

The relation at the end of subsection 46.D and Lemmas 46.21, 46.22, and 46.23 show that θ is a well-defined homomorphism. Formulas (46.24) and (46.25) yield

Proposition 46.26. The maps π and θ are isomorphisms inverse to each other.

Corollary 46.27. The group A(Q)/A′ is generated by products ãb̃c̃ for all ordered triples a, b, c of elements in Q^{×} satisfying abc = 1 with the following set of defining relations:

(R1′): (ãb̃(c̃d))·((ãb)c̃d̃) = (b̃c̃(d̃a))·((d̃a)b̃c̃) for all a, b, c, d ∈ Q^{×} such that abcd = 1;

(R2′): ãb̃c̃ = 1 if a and b commute.

46.F. Proof of Proposition 46.6. To prove Proposition 46.6, we need to prove that the homomorphism ∂_{1} in (46.10) is an isomorphism.

We shall view the fraction l_{a}/l_{b} for a, b ∈ Q \ F as a nonzero rational function on C, i.e., l_{a}/l_{b} ∈ F (C)^{×}.

Lemma 46.28. Let K_{0} be the quadratic subfield of Q corresponding to the fixed closed point x_{0} of degree 2 on C and let b ∈ K_{0} \ F. Then the space L_{1} consists of all the fractions l_{a}/l_{b} with a ∈ Q.

Proof. Obviously, l_{a}/l_{b} ∈ L_{1}. It follows from Lemma 45.5 that the space of all fractions l_{a}/l_{b} is 3-dimensional. On the other hand, dim L_{1} = 3. □

By Lemma 46.28, the group M′ is generated by symbols of the form {l_{a}/l_{b}, α} for all a, b ∈ Q \ F and α ∈ F^{×} and the group M_{1} is generated by symbols {l_{a}/l_{b}, l_{c}/l_{d}} for all a, b, c, d ∈ Q \ F.

Let a, b, c ∈ Q satisfy abc = 1. We define an element

[a, b, c] ∈ M_{1}/M′

as follows: If at least one of a, b and c belongs to F^{×}, we set [a, b, c] = 0. Otherwise, the linear forms l_{a}, l_{b} and l_{c} are nonzero and we set

[a, b, c] := \{\frac{l_{a}}{l_{c}} , \frac{l_{b}}{l_{c}} \} + M′.

Lemma 45.5 and the equality {u, −u} = 0 in K_{2}(F (C)) yield:

Lemma 46.29. Let a, b, c ∈ Q^{×} be such that abc = 1 and let α ∈ F^{×}. Then (1) [a, b, c] = [b, c, a]; (2) [αa, α^{−1}b, c] = [a, b, c];

(3) [a, b, c] + [c^{−1}, b^{−1}, a^{−1}] = 0;

(4) If a and b commute, then [a, b, c] = 0.

Lemma 46.30. ∂_{1}([a, b, c]) = ãb̃c̃.

Proof. We may assume that none of a, b, or c is a constant. Let x, y, and z be the points of C of degree 2 corresponding to quadratic subfields F [a], F [b], and F [c] that we identify with F (x), F (y) and F (z), respectively.

Consider the following element in the class [a, b, c]:

w = \{\frac{l_{a}}{l_{c}} , \frac{l_{b}}{l_{c}} \} + \{\frac{l_{b}}{l_{c}} , Nrd(a)\} + \{\frac{l_{b}}{l_{a}} , − Nrd(b)\}.

By Proposition 45.12 (identifying residue fields with the corresponding quadratic extensions) and Lemma 45.5,

∂_{x}(w) = \frac{l_{b}}{l_{c}} (x)(− Nrd(b))^{−1} = − Nrd(b)\frac{l_{b^{−1}}}{l_{b^{−1}a^{−1}}} (x)(− Nrd(b))^{−1} = a,
∂_{y}(w) = \frac{l_{c}}{l_{a}} (y)(− Nrd(ab)) = − Nrd(a)^{−1} \frac{l_{b^{−1}a^{−1}}}{l_{a^{−1}}} (x)(− Nrd(ab))
= − Nrd(a)^{−1}b̄^{−1}(− Nrd(ab)) = b,
∂_{z}(w) = −\frac{l_{a}}{l_{b}} (z) Nrd(a)^{−1} = Nrd(a)\frac{l_{bc}}{l_{b}} (x) Nrd(a)^{−1} = c.  □

Lemma 46.31. Let a, b, c, d ∈ Q \ F be such that cd, da ∉ F and abcd = 1. Then

\{\frac{l_{a}l_{c}}{l_{cd}l_{da}} , \frac{l_{b}l_{d}}{l_{cd}l_{da}} \} ∈ M′.

Proof. In Proposition 45.8 plugging in the elements c^{−1}, ab and b for a, b and c, respectively, and using Lemma 45.5, we get elements α, β, γ ∈ F^{×} such that on the conic C,

αl_{a}l_{c} + βl_{b}l_{d} + γl_{cd}l_{da} = 0.

Then

−\frac{αl_{a}l_{c}}{γl_{cd}l_{da}} − \frac{βl_{b}l_{d}}{γl_{cd}l_{da}} = 1

and

0 = \{ − \frac{αl_{a}l_{c}}{γl_{cd}l_{da}} , −\frac{βl_{b}l_{d}}{γl_{cd}l_{da}} \} ≡ \{\frac{l_{a}l_{c}}{l_{cd}l_{da}} , \frac{l_{b}l_{d}}{l_{cd}l_{da}} \} mod M′.  □

Proposition 46.32. Let a, b, c, d ∈ Q^{×} be such that abcd = 1. Then

[a, b, cd] + [ab, c, d] = [b, c, da] + [bc, d, a].

Proof. We first note that if one of the elements a, b, ab, c, d, cd belongs to F^{×}, the equality holds. For example, if a ∈ F^{×}, then the equality reads [ab, c, d] = [b, c, da] and follows from Lemma 46.29 and if α = ab ∈ F^{×}, then again by Lemma 46.29,

L.H.S. = 0 = [b, c, da] + [(da)^{−1}, α^{−1}c^{−1}, αb^{−1}] = R.H.S.

So we may assume that none of the elements belong to F^{×}. It follows from Lemma 45.5(4) that l_{cd}/l_{ab} and l_{da}/l_{bc} belong to F^{×}. By Lemmas 46.29 and 46.31, we have in M_{1}/M′:

0 = \{\frac{l_{a}l_{c}}{l_{cd}l_{da}} , \frac{l_{b}l_{d}}{l_{cd}l_{da}} \} + M′

= \{\frac{l_{a}}{l_{cd}} , \frac{l_{b}}{l_{cd}} \} + \{\frac{l_{c}}{l_{da}} , \frac{l_{b}}{l_{da}} \} + \{\frac{l_{a}}{l_{cd}} , \frac{l_{d}}{l_{da}} \} + \{\frac{l_{c}}{l_{da}} , \frac{l_{d}}{l_{cd}} \} + M′

= [a, b, cd] − [b, c, da] + (\{\frac{l_{a}}{l_{da}} , \frac{l_{d}}{l_{da}} \} + \{\frac{l_{da}}{l_{cd}} , \frac{l_{d}}{l_{da}} \}) + \{\frac{l_{c}}{l_{da}} , \frac{l_{d}}{l_{cd}} \} + M′

= [a, b, cd] − [b, c, da] + \{\frac{l_{a}}{l_{bc}} , \frac{l_{d}}{l_{bc}} \} + (\{\frac{l_{da}}{l_{cd}} , \frac{l_{d}}{l_{cd}} \} + \{\frac{l_{c}}{l_{da}} , \frac{l_{d}}{l_{cd}} \}) + M′

= [a, b, cd] − [b, c, da] − [bc, d, a] + \{\frac{l_{c}}{l_{cd}} , \frac{l_{d}}{l_{cd}} \} + M′

= [a, b, cd] − [b, c, da] − [bc, d, a] + \{\frac{l_{c}}{l_{ab}} , \frac{l_{d}}{l_{ab}} \} + M′

= [a, b, cd] − [b, c, da] − [bc, d, a] + [ab, c, d].  □

We shall use the presentation of the group A(Q)/A′ by generators and relations given in Corollary 46.27. We define a homomorphism

μ : A(Q)/A′ → M_{1}/M′

by the formula

μ(ãb̃c̃) := [a, b, c]

for all a, b, c ∈ Q such that abc = 1. It follows from Lemma 46.29(4) and Proposition 46.32 that μ is well-defined. Lemma 46.30 implies that ∂_{1} ∘ μ is the identity.

To show that μ is the inverse of ∂_{1} it suffices to prove that μ is surjective. The group M_{1}/M′ is generated by elements of the form w = {l_{a′}/l_{c′}, l_{b′}/l_{c′}} +

M′ for a′, b′, c′ ∈ Q\F. We may assume that 1, a′, b′ and c′ are linearly independent (otherwise, w = 0). In particular, 1, a′, b′ and a′b′ form a basis of Q, hence

c′ = α + βa′ + γb′ + δa′b′

for some α, β, γ, δ ∈ F with δ ≠ 0. We have

(γδ^{−1} + a′)(β + δb′) = ε + c′

for ε = βγδ^{−1} − α. Set

a := γδ^{−1} + a′, b := β + δb′, c := (ε + c′)^{−1}.

We have abc = 1. It follows from Lemma 45.5 that

w = \{\frac{l_{a′}}{l_{c′}} , \frac{l_{b′}}{l_{c′}} \} + M′ = \{\frac{l_{a}}{l_{c}} , \frac{l_{b}}{l_{c}} \} + M′ = [a, b, c].

By definition of μ, we have μ(ãb̃c̃) = [a, b, c] = w, hence μ is surjective. The proof of Proposition 46.6 is complete. □

47. Hilbert Theorem 90 for K_{2}

In this section we prove the K_{2}-analog of the classical Hilbert Theorem 90. Throughout this section let L/F be a Galois quadratic field extension with the

Galois group G = {1, σ}. For every field extension E/F linearly disjoint from L/F, the field LE = L ⊗_{F} E is a quadratic Galois extension of E with Galois group

47. HILBERT THEOREM 90 FOR K_{2} 209

isomorphic to G. The group G acts naturally on K_{2}(LE). We write (1 − σ)u for u − σ(u) with u ∈ K_{2}(LE) and set

V (E) := K_{2}(LE)/(1 − σ)K_{2}(LE).

If E → E′ is a homomorphism of field extensions of F linearly disjoint from L/F, there is a natural homomorphism

V (E) → V (E′).

Proposition 47.1. Let C be a conic curve over F and L/F a Galois quadratic field extension such that C has a rational point over L. Then the natural homomorphism V (F) → V (F (C)) is injective.

Proof. Let u ∈ K_{2}(L) satisfy u_{L(C)} = (1 − σ)v for some v ∈ K_{2}(L(C)). For a closed point x ∈ C the L-algebra L(x) := L ⊗_{F} F (x) is isomorphic to the direct product of residue fields L(y) for all closed points y ∈ C_{L} over x ∈ C. We denote the product of all the ∂_{y}(v) ∈ L(y)^{×} with y over x by ∂_{x}(v) ∈ L(x)^{×}.

Set a_{x} := ∂_{x}(v) ∈ L(x)^{×}. Then

a_{x}/σ(a_{x}) = ∂_{x}(v)/σ(∂_{x}(v)) = ∂_{x}((1 − σ)v) = ∂_{x}(u_{L(C)}) = 1,

i.e., a_{x} ∈ F (x)^{×}. Applying Theorem 46.1 to C_{L}, we have

∏_{x∈C} N_{F (x)/F} (a_{x}) = N_{L/F} ( ∏_{y∈C_{L}} N_{L(y)/L}(a_{y})) = N_{L/F} ( ∏_{y∈C_{L}} N_{L(y)/L}(∂_{y}(v))) = 1.

Hence applying Theorem 46.1 to C, there is a w ∈ K_{2}(F (C)) satisfying ∂_{x}(w) = a_{x} for all x ∈ C. Set v′ := v − w_{L(X)} ∈ K_{2}(L(C)). As

∂_{x}(v′) = ∂_{x}(v)∂_{x}(w)^{−1} = a_{x}a_{x}^{−1} = 1,

applying Theorem 46.1 to C_{L}, there exists an s ∈ K_{2}(L) with s_{L(C)} = v′. We have

(1 − σ)s_{L(C)} = (1 − σ)v′ = (1 − σ)v = u_{L(C)},

i.e., (1−σ)s−u splits over L(C). Since L(C)/L is a purely transcendental extension, we must have (1 − σ)s − u = 0 (cf. Example 100.6), hence

u = (1 − σ)s ∈ Im(1 − σ).  □

Corollary 47.2. For any finitely generated subgroup H ⊂ F^{×}, there is a field extension F′/F linearly disjoint from L/F such that the natural homomorphism V (F) → V (F′) is injective and H ⊂ N_{L′/F′}(L′^{×}) where L′ = LF′.

Proof. By induction it suffices to assume that H is generated by one element b. Set F′ = F (C), where C = C_{Q} is the conic curve associated with the quaternion algebra Q = (L/F, b) (cf. §98.E). Since Q splits over F′, we have b ∈ N_{L′/F′}(L′^{×}) by Facts 98.13(4) and 98.14(5). The conic C has a rational point over L, therefore, the homomorphism V (F) → V (F′) is injective by Proposition 47.1. □

For any two elements x, y ∈ L^{×}, we write ⟨x, y⟩ for the class of the symbol {x, y} in V (F). Let f be the group homomorphism

f = f_{F} : N_{L/F} (L^{×}) ⊗ F^{×} → V (F), f(N_{L/F} (x) ⊗ a) = ⟨x, a⟩.

The map f is well-defined. Indeed, if N_{L/F} (x) = N_{L/F} (y) for x, y ∈ L^{×}, then y = xzσ(z)^{−1} for some z ∈ L^{×} by the classical Hilbert Theorem 90. Hence {y, a} = {x, a} + (1 − σ){z, a} and consequently ⟨y, a⟩ = ⟨x, a⟩.

Lemma 47.3. Let b ∈ N_{L/F} (L^{×}). Then f(b ⊗ (1 − b)) = 0.

Proof. If b = d^{2} for some d ∈ F^{×}, then
f(b ⊗ (1 − b)) = ⟨d, 1 − d^{2}⟩ = ⟨d, 1 − d⟩ + ⟨d, 1 + d⟩ = ⟨−1, 1 + d⟩ = 0

since −1 = zσ(z)^{−1} for some z ∈ L^{×}.

So we may assume that b is not a square in F. Set

F′ = F [t]/(t^{2} − b), L′ = L[t]/(t^{2} − b).

Note that L′ is either a field or product of two copies of the field F′. Let u ∈ F′ be the class of t, so that u^{2} = b. Choose x ∈ L^{×} with N_{L/F} (x) = b. Note that N_{L′/F′}(x/u) = b/u^{2} = 1 and N_{L′/L}(1 − u) = 1 − b.

The automorphism σ extends to an automorphism of L′ over F′. Applying the classical Hilbert Theorem 90 to the extension L′/F′, there is a v ∈ L′^{×} such that vσ(v)^{−1} = x/u. We have

f(b⊗(1−b)) = ⟨x, 1−b⟩ = ⟨x, N_{L′/L}(1−u)⟩ = c_{F′/F} ⟨x, 1−u⟩ = c_{F′/F} (⟨x/u, 1−u⟩)

= c_{F′/F} (⟨vσ(v)^{−1}, 1 − u⟩) = (1 − σ)c_{F′/F} (⟨v, 1 − u⟩) = 0,

where c_{F′/F} : V (F′) → V (F) is induced by the norm map c_{L′/L} : K_{2}(L′) → K_{2}(L). □

Theorem 47.4 (Hilbert Theorem 90 for K_{2}). Let L/F be a Galois quadratic extension and σ the generator of Gal(L/F). Then the sequence

K_{2}(L) ⟶^{1−σ} K_{2}(L) ⟶^{c_{L/F}} K_{2}(F)

is exact.

Proof. Let u ∈ K_{2}(L) satisfy c_{L/F} (u) = 0. By Proposition 100.2, the group K_{2}(L) is generated by symbols of the form {x, a} with x ∈ L^{×} and a ∈ F^{×}. Therefore, we can write

u = ∑_{j=1}^{m}{x_{j}, a_{j}}

for some x_{j} ∈ L^{×} and a_{j} ∈ F^{×}, and

c_{L/F} (u) = ∑_{j=1}^{m}{N_{L/F} (x_{j}), a_{j}} = 0.

Hence by definition of K_{2}(F), we have in F^{×} ⊗ F^{×}:

(47.5)  ∑_{j=1}^{m} N_{L/F} (x_{j}) ⊗ a_{j} = ∑_{i=1}^{n} ± (b_{i} ⊗ (1 − b_{i}))

for some b_{i} ∈ F^{×}. Clearly, the equality (47.5) holds in H ⊗ F^{×} for some finitely generated subgroup H ⊂ F^{×} containing all the N_{L/F} (x_{j}) and b_{i}.

By Corollary 47.2, there is a field extension F′/F linearly disjoint from L/F such that the natural homomorphism V (F) → V (F′) is injective and

H ⊂ N_{L′/F′}(L′^{×})

with L′ = LF′. The equality (47.5) then holds in N_{L′/F′}(L′^{×}) ⊗ F′^{×}. Now apply the map f_{F′} to both sides of (47.5). By Lemma 47.3, the class of u_{L′} in V (F′) is equal to

∑_{j=1}^{m}⟨x_{j}, a_{j}⟩ = f_{F′}( ∑_{j=1}^{m} N_{L/F} (x_{j}) ⊗ a_{j}) = ∑_{i=1}^{n} ±f_{F′}(b_{i} ⊗ (1 − b_{i})) = 0,

i.e., u_{L′} ∈ (1 − σ)K_{2}(L′). Since the map V (F) → V (F′) is injective, we conclude that u ∈ (1 − σ)K_{2}(L). □

Theorem 47.6. Let u ∈ K_{2}(F) satisfy 2u = 0. Then u = {−1, a} for some a ∈ F^{×}. In particular, u = 0 if char(F) = 2.

Proof. Let G = {1, σ} be a cyclic group of order two. Consider a G-action on the field L = F ((t)) of a Laurent power series defined by

σ(t) = \{ \begin{matrix}−t & if char F ≠ 2, \\ t/(1 + t) & if char F = 2. \end{matrix}

We have a Galois quadratic extension L/E with E = L^{G}.

Consider the diagram

K_{2}(L) ⟶^{1−σ} K_{2}(L)
∂↓  ↓^{s}
F^{×} ⟶^{{−1}} K_{2}(F),

where ∂ is the residue homomorphism of the canonical discrete valuation of L, the map s = s_{t} is the specialization homomorphism of the parameter t (cf. §100.D), and the bottom homomorphism is multiplication by {−1}. We claim that the diagram is commutative. The group K_{2}(L) is generated by elements of the form {f, g} and {t, g} with f and g in F [[t]] having nonzero constant term. If char F ≠ 2, we have

s ∘ (1 − σ)({f, g}) = s({f, g} − {σ(f), σ(g)})
= {f(0), g(0)} − {σ(f)(0), σ(g)(0)}
= 0 = {−1} · ∂({f, g})

and

s ∘ (1 − σ)({t, g}) = s({t, g} − {−t, σ(g)})
= {−1, g(0)}
= {−1} · ∂({t, g}).

If char F = 2, we obviously have s(u) = s(σ(u)) for every u ∈ K_{2}(L), hence s ∘ (1 − σ) = 0.

Since c_{L/E}(u_{L}) = 2u_{E} = 0, by Theorem 47.4, we have u = (1 − σ)v for some v ∈ K_{2}(L). The commutativity of the diagram yields

u = s(u_{L}) = s((1 − σ)v) = {−1, ∂(v)}.  □

48. Proof of the main theorem

In this section we prove Theorem 44.1. Let F be a field of characteristic different from 2.

48.A. Injectivity of h_{F}. Suppose that h_{F} (u + 2K_{2}(F)) = 1 for an element u ∈ K_{2}(F). Let u be a sum of n symbols. We prove by induction on n that u ∈ 2K_{2}(F).

First consider the case n = 1, i.e., u = \{a, b\} with a, b ∈ F^{×}. Since \binom{a,b}{F} is a

split quaternion algebra, there are x, y ∈ F such that b = x^{2} − ay^{2}. It follows from Lemma 100.3 that {a, b} ∈ 2K_{2}(F).

Next consider the case n = 2, i.e. u = {a, b} + {c, d}. By assumption, the algebra \binom{a,b}{F} ⊗ \binom{c,d}{F} is split, or equivalently, \binom{a,b}{F} and \binom{c,d}{F} are isomorphic. By the Chain Lemma 98.15, we may assume that a = c and hence u = {a, bd}, so the statement follows from the case n = 1.

Now consider the general case. Write u in the form u = {a, b} + v for a, b ∈ F^{×} and an element v ∈ K_{2}F that is a sum of n − 1 symbols. We may assume that \{a, b\} ∉ 2K_{2}(F), i.e., the quaternion algebra Q := \binom{a,b}{F} does not split. Let C = C_{Q} be the conic curve over F corresponding to Q and set L = F (C). The conic C is given by the equation

at_{0}^{2} + bt_{1}^{2} = abt_{2}^{2}

in projective coordinates (cf. Example 45.3). Set x = t_{0}/t_{2} and y = t_{1}/t_{2}. As b^{−1}x^{2} + a^{−1}y^{2} = 1, we have

0 = {b^{−1}x^{2}, a^{−1}y^{2}} = 2{x, a^{−1}y^{2}} − 2{b, y} − {a, b}

and therefore {a, b} = 2r in K_{2}(L) with r = {x, a^{−1}y^{2}} − {b, y}. Let p ∈ C be the point of degree 2 given by Z = 0. The element r has only one nontrivial residue at the point p and ∂_{p}(r) = −1.

Since the quaternion algebra \binom{a,b}{F} is split over L, we have h_{L}(v_{L} +2K_{2}L) = 1. By induction, v_{L} = 2w for some element w ∈ K_{2}(L).

Set c_{x} := ∂_{x}(w) for every point x ∈ C. Since

c_{x}^{2} = ∂_{x}(2w) = ∂_{x}(v_{L}) = 1,

we have c_{x} = (−1)^{n_{x}} for n_{x} = 0 or 1. By Corollary 45.2(2), the degree of every point of C is even, hence

∑_{x∈C} n_{x} deg(x) = 2m

for some m ∈ ℤ. As every degree zero divisor on C is principal by Corollary 45.2(1), there is a function f ∈ L^{×} with div(f) = ∑ n_{x}x − mp. Set

w′ := w + {−1, f} + kr ∈ K_{2}(L)

where k = m + n_{p}. If x ∈ C is a point different from p, then

∂_{x}(w′) = ∂_{x}(w) · (−1)^{n_{x}} = 1.

Since, also,

∂_{p}(w′) = ∂_{p}(w) · (−1)^{m} · (−1)^{k} = (−1)^{n_{p}+m+k} = 1,

we have ∂_{x}(w′) = 1 for all x ∈ C. By Theorem 46.1, it follows that w′ = s_{L} for some s ∈ K_{2}(F). Thus

v_{L} = 2w = 2w′ − 2kr = 2s_{L} − {a^{k}, b}_{L}.

Set v′ := v − 2s + {a^{k}, b} ∈ K_{2}(F); we have v′_{L} = 0. The conic C has a rational point over the quadratic extension E = F (\sqrt{a}). Since the field extension E(C)/E is purely transcendental and v′_{E(C)} = 0, we have v′_{E} = 0 by Example 100.6 and therefore 2v′ = N_{E/F} (v′_{E}) = 0. It follows from Theorem 47.6 that v′ = {−1, d} for some d ∈ F^{×}. Hence modulo 2K_{2}(F), the element v is the sum of two symbols {a^{k}, b} and {−1, d}. Consequently, we are reduced to the case n = 2 that has already been considered.

48.B. Surjectivity of h_{F}. Recall that we are writing k_{2}(F) for K_{2}F/2K_{2}(F).

Proposition 48.1. Let L/F be a quadratic extension. Then the sequence

k_{2}(F) ⟶^{r_{L/F}} k_{2}(L) ⟶^{c_{L/F}} k_{2}(F)

is exact.

Proof. Let u ∈ K_{2}(L) satisfy c_{L/F} (u) = 2v for some v ∈ K_{2}(F). Then
c_{L/F} (u − v_{L}) = 2v − 2v = 0

and, by Theorem 47.4, we have u − v_{L} = (1 − σ)w for some w ∈ K_{2}(L). Hence

u = v_{L} + (1 − σ)w = (v + c_{L/F} (w))_{L} − 2σw.  □

We now finish the proof of Theorem 44.1. Let s ∈ Br_{2}(F). First suppose that the field F is 2-special (cf. §101.B). By induction on the index of s, we prove that s ∈ Im(h_{F}). By Proposition 101.15, there exists a quadratic extension L/F with ind(s_{L}) < ind(s). By induction, s_{L} = h_{L}(u) for some u ∈ k_{2}(L). It follows from Proposition 101.9 that

h_{F} (c_{L/F} (u)) = c_{L/F} (h_{L}(u)) = c_{L/F} (s_{L}) = 1.

The injectivity of h_{F} implies that c_{L/F} (u) = 0 so by Proposition 48.1, we have u = v_{L} for some v ∈ k_{2}(F). Therefore,

h_{F} (v)_{L} = h_{L}(v_{L}) = h_{L}(u) = s_{L},

hence s−h_{F} (v) splits over L and must be the class of a quaternion algebra. Consequently, s−h_{F} (v) = h_{F} (w) for some symbol w in k_{2}(F), so s = h_{F} (v+w) ∈ Im(h_{F}).

In the general case, applying the first part of the proof to a maximal odd degree extension of F (cf. §101.B and Proposition 101.16), we see that there exists an odd degree extension E/F such that s_{E} = h_{E}(v) for some v ∈ k_{2}(E). Then again by Proposition 101.9,

s = c_{E/F} (s_{E}) = c_{E/F} (h_{E}(v)) = h_{F} (c_{E/F} (v)).

Part 2

Algebraic cycles

CHAPTER IX Homology and Cohomology

The word “scheme” in this book (with the exception of §49) will mean a separated scheme of finite type over a field and a “variety” will mean an integral scheme.

In this chapter we develop the K-homology and K-cohomology theories of schemes over a field that generalizes the theory of Chow groups. We follow the approach of [117] given by Rost. There are two advantages of having such general theories rather than just that of Chow groups. First we have a long (infinite) localization exact sequence. This tool together with the 5-lemma allows us to give simple proofs of some basic results in the theory such as the Homotopy Invariance and Projective Bundle Theorems. Secondly, the construction of the deformation map (called the specialization homomorphism in [45]), used in the definition of the pull-back homomorphisms, is much easier; it does not require intersections with Cartier divisors.

We view the K-homology as a covariant functor from the category of schemes of finite type over a field to the category of abelian groups and the K-cohomology as a contravariant functor from the category of smooth schemes of finite type over a field. The fact that K-homology groups for smooth schemes coincide with Kcohomology groups should be viewed as Poincaré duality.

49. The complex C_{∗}(X)

The purpose of this section is to construct complexes C_{∗}(X) giving the homology and cohomology theories that we need.

Throughout this section (and only in this section), we need to extend the class of separated schemes of finite type over a field and consider the class of excellent schemes (cf. [48, §7.8]). The class of excellent schemes contains:

1. Schemes of finite type over a field.

2. Closed and open subschemes of excellent schemes.

3. Spec(O_{X,x}) where x is a point of a scheme X of finite type over a field.

4. Spec(R) where R is a complete noetherian local ring.

We shall use the following properties of excellent schemes:

A. If X is excellent integral, then the normalization morphism X̃ → X is finite and X̃ is excellent.

B. An excellent scheme X is catenary, i.e., given irreducible closed subschemes Z ⊂ Y ⊂ X, all maximal chains of closed irreducible subsets between Z and Y have the same length.

If x is a point of a scheme X, we write κ(x) for the residue field of x (and we shall use the standard notation F (x) when X is a scheme over a field F). We write dim x for the dimension of the closure \overline{\{x\}} and X_{(p)} for the set of point of X of dimension p.

The word “scheme” in this section will mean an excellent scheme of finite dimension.

49.A. Residue homomorphism for local rings. Let R be a 1-dimensional local excellent domain with quotient field L and residue field E. Let R̃ denote the integral closure of R in L. The ring R̃ is semilocal, 1-dimensional, and finite as Ralgebra. Let M_{1}, M_{2}, …, M_{n} be all of the maximal ideals of R̃. Each localization R̃_{M_{i}} is integrally closed, noetherian and 1-dimensional, hence a DVR. Let v_{i} denote the discrete valuation of R̃_{M_{i}} and E_{i} its residue field. The field extension E_{i}/E is finite. Let K_{∗} denote the Milnor K-groups (cf. §100) and define the residue homomorphism

∂_{R} : K_{∗}(L) → K_{∗−1}(E),

by the formula

∂_{R} = ∑_{i=1}^{n} c_{E_{i}/E} ∘ ∂_{v_{i}},

where

∂_{v_{i}} : K_{∗}(L) → K_{∗−1}(E_{i})

is the residue homomorphism associated with the discrete valuation v_{i} on L (cf. §100.B) and

c_{E_{i}/E} : K_{∗−1}(E_{i}) → K_{∗−1}(E)

is the norm homomorphism (cf. §100.E).

Let X be a scheme. For every pair of points x, x′ ∈ X, we define a homomorphism

∂_{x′}^{x} : K_{∗}(κ(x)) → K_{∗−1}(κ(x′))

as follows: Let Z be the closure of {x} in X considered as an integral closed subscheme (subvariety) of X. If x′ ∈ Z (in this case we say that x′ is a specialization of x) and dim x = dim x′ + 1, then the local ring R = O_{Z,x′} is a 1-dimensional excellent local domain with quotient field κ(x) and residue field κ(x′). We set ∂_{x′}^{x} = ∂_{R}. Otherwise, set ∂_{x′}^{x} = 0.

Lemma 49.1. Let X be a scheme. For each x ∈ X and every α ∈ K_{∗}(κ(x)) the residue ∂_{x′}^{x}(α) is nontrivial for only finitely many points x′ ∈ X.

Proof. We may assume that X = Spec(A) where A is an integrally closed noetherian domain, x the generic point of X and α the symbol {a_{1}, a_{2}, …, a_{n}} with nonzero a_{i} ∈ A. For every point x′ ∈ X of codimension 1, let v_{x′} be the corresponding discrete valuation of the quotient field of A. For each i, there is a bijection between the set of all x′ satisfying v_{x′}(a_{i}) ≠ 0 and the set of minimal prime ideals of the (noetherian) ring A/a_{i}A and hence this set is finite. Thus, for all but finitely many x′ we have v_{x′}(a_{i}) = 0 for all i and therefore ∂_{x′}^{x}(α) = 0. □

It follows from Lemma 49.1 that for a scheme X there is a well-defined endomorphism d = d_{X} of the direct sum

C(X) := ∐_{x∈X} K_{∗}(κ(x))

such that the (x, x′)-component of d is equal to ∂_{x′}^{x}.

49. THE COMPLEX C_{∗}(X) 219

Example 49.2. Let Z ⊂ X be a closed subscheme and let z_{1}, z_{2}, … be all of the generic points of Z. We set

[Z] := ∑ m_{i}z_{i} ∈ ∐_{x∈X} K_{0}(κ(x)) ⊂ C(X),

where m_{i} = l(O_{Z,z_{i}}) is the length of the local ring O_{Z,z_{i}}. The element [Z] is called the cycle of Z on X.

Example 49.3. Let X be a scheme, x ∈ X, and f ∈ κ(x)^{×}. We view f as an element of K_{1}(κ(x)) ⊂ C(X). Then the element

d_{X}(f) ∈ ∐_{x∈X} K_{0}(κ(x)) ⊂ C(X)

is called the divisor of f and is denoted by div(f) .

The group C(X) is graded: we write for any p ≥ 0,

C_{p}(X) := ∐_{x∈X_{(p)}} K_{∗}(κ(x)).

The endomorphism d_{X} of C_{∗}(X) has degree −1 with respect to this grading. We also set

C_{p,n}(X) := ∐_{x∈X_{(p)}} K_{p+n}(κ(x)),

hence C_{p}(X) is the coproduct of C_{p,n}(X) over all n. Note that the graded group C_{∗,n}(X) is invariant under d_{X} for every n.

Let X be a scheme over a field F. Then the group C_{p}(X) has a natural structure of a left and right K_{∗}(F)-module for all p and d_{X} is a homomorphism of right K_{∗}(F)-modules.

If X is the disjoint union of two schemes X_{1} and X_{2}, we have

(49.4)  C_{∗}(X) = C_{∗}(X_{1}) ⊕ C_{∗}(X_{2})

and d_{X} = d_{X_{1}} ⊕ d_{X_{2}}.

49.B. Multiplication with an invertible function. Let a be an invertible regular function on a scheme X. For every α ∈ C_{∗}(X), we write {a} · α for the element of C_{∗}(X) satisfying

({a} · α)_{x} = {a(x)} · α_{x}

for every x ∈ X. We denote by {a} the endomorphism of C_{∗}(X) given by α ↦ {a} · α.

The product α · {a} is defined similarly.

Let a_{1}, a_{2}, …, a_{n} be invertible regular functions on a scheme X. We write {a_{1}, a_{2}, …, a_{n}} · α for the product {a_{1}} · {a_{2}} · . . . · {a_{n}} · α and {a_{1}, a_{2}, …, a_{n}} for the endomorphism of C_{∗}(X) given by α ↦ {a_{1}, a_{2}, …, a_{n}} · α.

Proposition 49.5. Let a be an invertible function on a scheme X and α ∈ C_{∗}(X). Then

d_{X}(α · {a}) = d_{X}(α) · {a} and d_{X}({a} · α) = −{a} · d_{X}(α).

Proof. The statement follows from Fact 100.4(1) and the projection formula for the norm map in Proposition 100.8(3). □

By Fact 100.1, it follows that

{a_{1}, a_{2}} = −{a_{2}, a_{1}} and {a_{1}, a_{2}} = 0 if a_{1} + a_{2} = 1.

49.C. Push-forward homomorphisms. Let f : X → Y be a morphism of schemes. We define the push-forward homomorphism

f_{∗} : C_{∗}(X) → C_{∗}(Y)

as follows: Let x ∈ X and y ∈ Y. If y = f(x) ∈ Y and the field extension κ(x)/κ(y) is finite, we set

(f_{∗})_{y}^{x} := c_{κ(x)/κ(y)} : K_{∗}(κ(x)) → K_{∗}(κ(y))

and set (f_{∗})_{y}^{x} = 0 otherwise. It follows from transitivity of the norm map that if g : Y → Z is another morphism, then (g ∘ f)_{∗} = g_{∗} ∘ f_{∗}.

If either

(1) f is a morphism of schemes of finite type over a field or

(2) f is a finite morphism, the push-forward f_{∗} is a graded homomorphism of degree 0. Indeed, if y = f(x), then dim y = dim x if and only if κ(x)/κ(y) is a finite extension for all x ∈ X.

If f is a morphism of schemes over a field F, then f_{∗} is a homomorphism of left and right K_{∗}(F)-modules.

Example 49.6. If f : X → Y is a closed embedding, then f_{∗} is a monomorphism satisfying f_{∗} ∘d_{X} = d_{Y} ∘f_{∗}. Moreover, if in addition f is a bijection on points (e.g., if f is the canonical morphism Y_{red} → Y), then f_{∗} is an isomorphism.

Remark 49.7. Let X be a localization of a scheme Y (e.g., X is an open subscheme of Y) and f : X → Y the natural morphism. For every point x ∈ X, the natural ring homomorphism O_{Y,f(x)} → O_{X,x} is an isomorphism. It follows from the definitions that for any x, x′ ∈ X, we have

(f_{∗} ∘ d_{X})_{y′}^{x} = (f_{∗})_{y′}^{x′} ∘ (d_{X})_{x′}^{x} = (d_{Y})_{y′}^{y} ∘ (f_{∗})_{y}^{x} = (d_{Y} ∘ f_{∗})_{y′}^{x}

where y = f(x) and y′ = f(x′). Note that if y′′ ∈ Y does not belong to the image of f, then (f_{∗} ∘ d_{X})_{y′′}^{x} = 0, but in general (d_{Y} ∘ f_{∗})_{y′′}^{x} may be nonzero.

The following rule is a consequence of the projection formula for Milnor’s Kgroups (cf. Fact 100.8(3)).

Proposition 49.8. Let f : X → Y be a morphism of schemes and a an invertible regular function on Y. Then

f_{∗} ∘ {a′} = {a} ∘ f_{∗}

where a′ = f^{∗}(a) = a ∘ f.

Proposition 49.9. Let f : X → Y be either (1) a proper morphism of schemes of finite type over a field or

(2) a finite morphism. Then the diagram

C_{p}(X) ⟶^{d_{X}} C_{p−1}(X)
f_{∗}↓  ↓^{f_{∗}}
C_{p}(Y) ⟶^{d_{Y}} C_{p−1}(Y)

is commutative.

Proof. Let x ∈ X_{(p)} and y′ ∈ Y_{(p−1)}. The (x, y′)-component of both compositions in the diagram can be nontrivial only if y′ belongs to the closure of the point y = f(x), i.e., if y′ is a specialization of y. We have

p = dim x ≥ dim y ≥ dim y′ = p − 1,

therefore, dim y can be either equal to p or p − 1. Note that if f is finite, then dim y = p. Case 1: dim(y) = p.

In this case, the field extension κ(x)/κ(y) is finite. Replacing X by the closure of {x} and Y by the closure of {y} we may assume that x and y are the generic points of X and Y respectively.

First suppose that X and Y are normal. Since the morphism f is proper, the points x′ ∈ X_{p−1} satisfying f(x′) = y′ are in a bijective correspondence with the extensions of the valuation v_{y′} of the field κ(y) to the field κ(x). Hence by Fact 100.8(4),

(d_{Y} ∘ f_{∗})_{y′}^{x} = ∂_{y′}^{y} ∘ c_{κ(x)/κ(y)}
= ∑_{f(x′)=y′} c_{κ(x′)/κ(y′)} ∘ ∂_{x′}^{x}
= ∑_{f(x′)=y′}(f_{∗})_{y′}^{x′} ∘ (d_{X})_{x′}^{x}
= (f_{∗} ∘ d_{X})_{y′}^{x}.

In the general case let g : X̃ → X and h : Ỹ → Y be the normalizations and let x̃ and ỹ be the generic points of X̃ and Ỹ respectively. Note that κ(x̃) ≃ κ(x) and κ(ỹ) ≃ κ(y). There is a natural morphism f̃: X̃ → Ỹ over f.

Consider the following diagram:

\xymatrix{K_{∗}(κ(x̃)) \ar[dd]^{c_{κ(x̃)/κ(ỹ)}} \ar[drr]^{∼} \ar[r]^{dX̃} & C_{p−1}(X̃) \ar[dd]_{f̃_{∗}} \ar[drr]^{g_{∗}} \\  &  & K_{∗}(κ(x)) \ar[dd]^{c_{κ(x)/κ(y)}} \ar[r]^{d_{X}} & C_{p−1}(X) \ar[dd]_{f_{∗}} \\ K_{∗}(κ(ỹ)) \ar[drr]^{∼} \ar[r]^{dỸ} & C_{p−1}(Ỹ) \ar[drr]^{h_{∗}} \\  &  & K_{∗}(κ(y)) \ar[r]^{d_{Y}} & C_{p−1}(Y)}

By the first part of the proof, the back face of the diagram is commutative. The left face is obviously commutative. The right face is commutative by functoriality of the push-forward. The upper and the bottom faces are commutative by definition of the maps d_{X} and d_{Y}. Hence the front face is also commutative, i.e., the (x, y′)- components of the compositions f_{∗} ∘ d_{X} and d_{Y} ∘ f_{∗} coincide.

Note that we have proved the proposition in the case when f is finite. Before proceeding to Case 2, as a corollary we also deduce Theorem 49.10 (Weil’s Reciprocity Law). Let X be a complete integral curve over a field F. Then the composition

K_{∗+1}(F (X)) ⟶^{d_{X}} ∐_{x∈X_{(0)}} K_{∗}(F (x)) \underrightarrow{∑ c_{κ(x)/F}} K_{∗}(F)

is trivial.

Proof. The case X = ℙ_{F}^{1} follows from Theorem 100.7. The general case can be reduced to the case of the projective line as follows. Let f be a nonconstant rational function on X. We view f as a finite morphism f : X → ℙ_{F}^{1} over F. By the first case of the proof of Proposition 49.9, the left square of the diagram

K_{∗+1}(F (X)) ⟶^{d_{X}} ∐_{x∈X_{(0)}} K_{∗}(F (x)) \underrightarrow{∑ c_{κ(x)/F}} K_{∗}(F)
↓^{f_{∗}}  ↓^{f_{∗}}  ∥
K_{∗+1}(F (ℙ^{1})) ⟶^{d_{ℙ^{1}}} ∐_{y∈ℙ_{(0)}^{1}} K_{∗}(F (y)) \underrightarrow{∑ c_{κ(y)/F}} K_{∗}(F)

is commutative. The right square is commutative by the transitivity property of the norm map (cf. Fact 100.8(1)). Finally, the statement of the theorem follows from the commutativity of the diagram. □

Weil’s Reciprocity Law can be reformulated as follows:

Corollary 49.11. Proposition 49.9 holds for the structure morphism X→ Spec(F).

We return to the proof of Proposition 49.9. Case 2: dim(y) = p − 1.

In this case y′ = y. We replace Y by Spec(κ(y)) and X by the fiber X ×_{Y} Spec(κ(y)) of f over y. We can further replace X by the closure of x in X. Thus, X is a proper integral curve over the field κ(y) and the result follows from Corollary 49.11. □

49.D. Pull-back homomorphisms. Let g : Y → X be a flat morphism of schemes. We say that g is of relative dimension d if for every x ∈ X in the image of g and for every generic point y of g^{−1}(\overline{\{x\}}) we have dim y = dim x + d.

In this book all flat morphisms will be assumed of constant relative dimension. Let g : Y → X be a flat morphism of relative dimension d. For every point

x ∈ X, let Y_{x} denote the fiber scheme

Y ×_{X} Spec(κ(x))

over κ(x). We identify the underlying topological space of Y_{x} with a subspace of Y.

The following statement is a direct consequence of the going-down theorem [99, Ch. 1, Th. 4].

Lemma 49.12. For every x ∈ X we have:

(1) dim y ≤ dim x + d for every y ∈ Y_{x}.

(2) A point y ∈ Y_{x} is generic in Y_{x} if and only if dim y = dim x + d.

If y is a generic point of Y_{x}, the local ring O_{Y_{x},y} is noetherian 0-dimensional, hence is artinian. We define the ramification index of y by

e_{y}(g) := l(O_{Y_{x},y}),

where l denotes the length (cf. §102).

The pull-back homomorphism

g^{∗} : C_{∗}(X) → C_{∗+d}(Y)

is defined as follows: Let x ∈ X and y ∈ Y. If g(y) = x and y is a generic point of Y_{x}, we set

(g^{∗})_{y}^{x} := e_{y}(g) · r_{κ(y)/κ(x)} : K_{∗}(κ(x)) → K_{∗}(κ(y))

where r_{κ(y)/κ(x)} is the restriction homomorphism (cf. §100.A) and (g_{∗})_{y}^{x} = 0 otherwise.

Suppose that Z is of pure dimension d over a field F. The structure morphism p : Z → Spec(F) is flat of relative dimension d. The image of the identity under the composition

p^{∗} : ℤ = K_{0}(F) = C_{0,0}(Spec F) ⟶^{p^{∗}} C_{d,−d}(Z) ⟶^{i_{∗}} C_{d,−d}(X),

where i : Z → X is the closed embedding, is equal to the cycle [Z] of Z.

Example 49.13. Let p : E → X be a vector bundle of rank r. Then p is a flat morphism of relative dimension r and p^{∗}([X]) = [E].

Example 49.14. Let X be a scheme of finite type over F and let L/F be an arbitrary field extension. The natural morphism g : X_{L} → X is flat of relative dimension 0. The pull-back homomorphism

g^{∗} : C_{p}(X) → C_{p}(X_{L})

is called the change of field homomorphism.

Example 49.15. An open embedding j : U → X is a flat morphism of relative dimension 0. The pull-back homomorphism

j^{∗} : C_{p}(X) → C_{p}(U)

is called the restriction homomorphism.

The following proposition is an immediate consequence of the definitions.

Proposition 49.16. Let g : Y → X be a flat morphism and a an invertible function on X. Then

g^{∗} ∘ {a} = {a′} ∘ g^{∗},

where a′ = g^{∗}(a) = a ∘ g.

Let g be a morphism of schemes over a field F. It follows from Proposition 49.16 that g^{∗} is a homomorphism of left and right K_{∗}(F)-modules.

Let g : Y → X and h : Z → Y be flat morphisms. Let z ∈ Z and y = h(z), x = g(y). It follows from Lemma 49.12 that z is a generic point of Z_{x} if and only if z is a generic point of Z_{y} and y is a generic point of Y_{x}.

Lemma 49.17. Let z be a generic point of Z_{x}. Then e_{z}(g ∘ h) = e_{z}(h) · e_{y}(g).

Proof. The statement follows from Corollary 102.2 with B = O_{Y_{x},y} and C = O_{Z_{x},z}. Note that C/QC = O_{Z_{y},z} where Q is the maximal ideal of B. □

Proposition 49.18. Let g : Y → X and h : Z → Y be flat morphisms of constant relative dimension. Then (g ∘ h)^{∗} = h^{∗} ∘ g^{∗}.

Proof. Let x ∈ X and z ∈ Z. We compute the (z, x)-components of both sides of the equality. We may assume that x = (g ∘h)(z). Let y = h(z). By Lemma 49.17, we have

((g ∘ h)^{∗})_{z}^{x} = e_{z}(g ∘ h) · r_{κ(z)/κ(x)}
= e_{z}(h) · e_{y}(g) · r_{κ(z)/κ(y)} ∘ r_{κ(y)/κ(x)}
= (h^{∗})_{z}^{y} ∘ (g^{∗})_{y}^{x}
= (h^{∗} ∘ g^{∗})_{z}^{x}.  □

Next consider the fiber product diagram

(49.19)  \begin{matrix}X′ & ⟶^{g′} & X \\ ^{f′}↓ & & ↓^{f} \\ Y′ & ⟶^{g} & Y. \end{matrix}

Proposition 49.20. Let g and g′ in (49.19) be flat morphisms of relative dimension d. Suppose that either

(1) f is a morphism of schemes of finite type over a field or

(2) f is a finite morphism. Then the diagram

C_{p}(X) ⟶^{g′^{∗}} C_{p+d}(X′)
f_{∗}↓  ↓^{f′_{∗}}
C_{p}(Y) ⟶^{g^{∗}} C_{p+d}(Y′)

is commutative.

Proof. Let x ∈ X_{(p)} and y′ ∈ Y′_{(p+d)}. We shall compare the (x, y′)-components of both compositions in the diagram. These components are trivial unless g(y′) = f(x). Denote this point by y. By Lemma 49.12,

p + d = dim y′ ≤ dim y + d ≤ dim x + d = p + d,

hence dim y = dim x = p and y′ is a generic point of Y′_{y}. In particular, the field extension κ(x)/κ(y) is finite.

Let S be the set of all x′ ∈ X′ satisfying f′(x′) = y′ and g′(x′) = x. Again by Lemma 49.12,

p + d = dim y′ ≤ dim x′ ≤ dim x + d = p + d,

hence dim x′ = dim y′ = p+d and x′ is a generic point of X′_{x}. In particular, the field extension κ(x′)/κ(y′) is finite. The set S is in a natural bijective correspondence with the finite set Spec(κ(y′) ⊗_{κ(y)} κ(x)).

The local ring C = O_{X′_{x},x′} is a localization of the ring O_{Y′_{y},y′} ⊗_{κ(y)} κ(x) and hence is flat over B = O_{Y′_{y},y′}. Let Q be the maximal ideal of B. The factor ring C/QC is the localization of the tensor product κ(y′) ⊗_{κ(y)} κ(x) at the prime ideal corresponding to x′. Denote by l_{x′} the length of C/QC.

By Corollary 102.2,

(49.21)  e_{x′}(g′) = l_{x′} · e_{y′}(g)

for every x′ ∈ S. It follows from (49.21) and Fact 100.8(5) that

(f′_{∗} ∘ g′^{∗})_{y′}^{x} = ∑_{x′∈S}(f′_{∗})_{y′}^{x′} ∘ (g′^{∗})_{x′}^{x}
= ∑_{x′∈S} e_{x′}(g′) · c_{κ(x′)/κ(y′)} ∘ r_{κ(x′)/κ(x)}
= e_{y′}(g) · ∑_{x′∈S} l_{x′} · c_{κ(x′)/κ(y′)} ∘ r_{κ(x′)/κ(x)}
= e_{y′}(g) · r_{κ(y′)/κ(y)} ∘ c_{κ(x)/κ(y)}
= (g^{∗})_{y′}^{y} ∘ (f_{∗})_{y}^{x}
= (g^{∗} ∘ f_{∗})_{y′}^{x}.  □

Remark 49.22. It follows from the definitions that Proposition 49.20 holds also for arbitrary f if Y′ is a localization of Y (cf. Remark 49.7).

Proposition 49.23. Let g : Y → X be a flat morphism of relative dimension d. Then the diagram

C_{p}(X) ⟶^{d_{X}} C_{p−1}(X)
g^{∗}↓  ↓^{g^{∗}}
C_{p+d}(Y) ⟶^{d_{Y}} C_{p+d−1}(Y)

is commutative.

Proof. Let x ∈ X_{(p)} and y′ ∈ Y_{(p+d−1)}. We compare the (x, y′)-components of both compositions in the diagram. Let y_{1}, …, y_{k} be all generic points of Y_{x} ⊂ Y satisfying y′ ∈ \overline{\{y_{i}\}}. We have

(49.24)  (d_{Y} ∘ g^{∗})_{y′}^{x} = ∑_{i=1}^{k}(d_{Y})_{y′}^{y_{i}} ∘ (g^{∗})_{y_{i}}^{x} = ∑_{i=1}^{k} e_{y_{i}}(g) · (d_{Y})_{y′}^{y_{i}} ∘ r_{κ(y_{i})/κ(x)}.

Set x′ = g(y′). If x′ ∉ \overline{\{x\}}, then both components (g^{∗} ∘ d_{X})_{y′}^{x} and (d_{Y} ∘ g^{∗})_{y′}^{x} are trivial.

Suppose x′ ∈ \overline{\{x\}}. We have

p = dim x ≥ dim x′ ≥ dim y′ − d = p − 1.

Therefore, dim x′ is either p or p − 1. Case 1: dim(x′) = p, i.e., x′ = x.

The component (g^{∗} ∘ d_{X})_{y′}^{x} is trivial since (g^{∗})x̃ ; y′ = 0 for every x̃ ≠ x′. By assumption, every discrete valuation of κ(y_{i}) with center y′ is trivial on κ(x). Therefore, the map (d_{Y})_{y′}^{y_{i}} is trivial on the image of r_{κ(y_{i})/κ(x)}. It follows from formula (49.24) that (d_{Y} ∘ g^{∗})_{y′}^{x} = 0. Case 2: dim(x′) = p − 1.

We have y′ is a generic point of Y_{x′} and

(49.25)  (g^{∗} ∘ d_{X})_{y′}^{x} = (g^{∗})_{y′}^{x′} ∘ (d_{X})_{x′}^{x} = e_{y′}(g) · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}.

Replacing X by \overline{\{x\}} and Y by g^{−1}(\overline{\{x\}}), we may assume that X = \overline{\{x\}}. By Propositions 49.9 and 49.20, we can replace X by its normalization X̃ and Y by the fiber product Y ×_{X} X̃, so we may assume that X is normal.

Let Y_{1}, …, Y_{k} be all irreducible components of Y containing y′, so that y_{i} is the generic point of Y_{i} for all i. Let Ỹ_{i} be the normalization of Y_{i} and let ỹ_{i} be the generic points of Ỹ_{i}. We have κ(ỹ_{i}) = κ(y_{i}). Let t be a prime element of the discrete valuation ring R = O_{X,x′}.

The local ring A = O_{Y,y′} is 1-dimensional; its minimal prime ideals are in bijective correspondence with the set of points y_{1}, …, y_{k}.

Fix i ∈ [1, k]. We write A_{i} for the factor ring of A by the corresponding minimal prime ideal. Since A is flat over R, the prime element t is not a zero divisor in A, hence the image of t in A_{i} is not zero for every i. Let Ã_{i} be the normalization of the ring A_{i}.

Let S_{i} be the set of all points w ∈ Ỹ_{i} such that g(w) = x′. There is a natural bijection between S_{i} and the set of all maximal ideals of Ã_{i}. Moreover, if Q is a maximal ideal of Ã_{i} corresponding to a point w ∈ S_{i}, then the local ring O_{Ỹ_{i},w} coincides with the localization of Ã_{i} with respect to Q.

Denote by l_{i,w} the length of the ring O_{Ỹ_{i},w}/tO_{Ỹ_{i},w}. Applying Lemma 102.3 to the A-algebra Ã_{i} and M = Ã_{i}/tÃ_{i}, we have

(49.26)  l_{A}(Ã_{i}/tÃ_{i}) = ∑_{w∈S_{i}} l_{i,w} · [κ(w) : κ(y′)].

On the other hand, l_{i,w} is the ramification index of the discrete valuation ring O_{Ỹ_{i},w} over R. It follows from Fact 100.4(2) that

(49.27)  ∂ỹ_{i} ; w ∘ r_{κ(y_{i})/κ(x)} = l_{i,w} · r_{κ(w)/κ(x′)} ∘ ∂_{x′}^{x}

for every w ∈ S_{i}.

By (49.26), (49.27), and Fact 100.8(3), for every i, we have

(d_{Y})_{y′}^{y_{i}} ∘ r_{κ(y_{i})/κ(x)} = ∑ c_{κ(w)/κ(y′)} ∘ ∂ỹ_{i} ; w ∘ r_{κ(y_{i})/κ(x)}
= ∑ c_{κ(w)/κ(y′)} · l_{i,w} · r_{κ(w)/κ(x′)} ∘ ∂_{x′}^{x}
= ∑ l_{i,w} · c_{κ(w)/κ(y′)} ∘ r_{κ(w)/κ(y′)} ∘ r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}
= ∑ l_{i,w} · [κ(w) : κ(y′)] · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}
= l_{A}(Ã_{i}/tÃ_{i}) · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}

(where all summations are taken over all w ∈ S_{i}).

The factor A-module Ã_{i}/A_{i} is of finite length, hence by Lemma 102.4, we have h(t, A_{i}) = h(t, Ã_{i}) where h is the Herbrand index. Since t is not a zero divisor in either A_{i} or in Ã_{i}, we have l_{A}(Ã_{i}/tÃ_{i}) = l_{A}(A_{i}/tA_{i}) = l(A_{i}/tA_{i}). Therefore,

(49.28)  (d_{Y})_{y′}^{y_{i}} ∘ r_{κ(y_{i})/κ(x)} = l(A_{i}/tA_{i}) · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}.

The local ring O_{Y_{x},y_{i}} = O_{Y,y_{i}} is the localization of A with respect to the minimal prime ideal corresponding to y_{i}. The ring O_{Y_{x′} ,y′} is canonically isomorphic to A/tA.

Applying Lemma 102.5 to the ring A and the module M = A we get the equality

(49.29) e_{y′}(g) = h(t, A) = ∑_{i=1}^{k} l(O_{Y_{x},y_{i}}) · l(A_{i}/tA_{i}) = ∑_{i=1}^{k} e_{y_{i}}(g) · l(A_{i}/tA_{i}).

It follows from (49.25), (49.28), and (49.29) that

(d_{Y} ∘ g^{∗})_{y′}^{x} = ∑_{i=1}^{k} e_{y_{i}}(g) · (d_{Y})_{y′}^{y_{i}} ∘ r_{κ(y_{i})/κ(x)}

= ∑_{i=1}^{k} e_{y_{i}}(g) · l(A_{i}/tA_{i}) · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}
= e_{y′}(g) · r_{κ(y′)/κ(x′)} ∘ ∂_{x′}^{x}
= (g^{∗} ∘ d_{X})_{y′}^{x}.  □

The following proposition was proven by Kato in [79].

Proposition 49.30. For every scheme X over a field, the map d_{X} is a differential of C_{∗}(X), i.e., (d_{X})^{2} = 0.

Proof. We will prove the statement in several steps. Step 1: X = Spec(R), where R = F [[s, t]] and F is a field.

A polynomial t^{n}+a_{1}t^{n−1}+a_{2}t^{n−2}+⋯+a_{n} over the ring F [[s]] is called marked if a_{i} ∈ sF [[s]] for all i. We shall use the following properties of marked polynomials derived from the Weierstrass Preparation Theorem [18, Ch.VII, §3, no. 8]:

A. Every height 1 ideal of the ring R is either equal to sR or is generated by a unique marked polynomial.

B. A marked polynomial f is irreducible in R if and only if f is irreducible in F ((s))[t].

It follows that the multiplicative group F ((s, t))^{×} is generated by R^{×}, s, t and the set H of all power series of the form t^{−n} · f where f is a marked polynomial of degree n.

If r ∈ R^{×} and α ∈ K_{∗}(F ((s, t))), then by Proposition 49.5,

(d_{X})^{2}({r} · α) = −d_{X}({r̄} · d_{X}(α)) = {r̄} · (d_{X})^{2}(α),

where r̄ ∈ F is the residue of r. Thus it suffices to prove the following: (i) (d_{X})^{2}({s, t}) = 0, (ii) (d_{X})^{2}({f, g_{1}, …, g_{n}}) = 0 where f ∈ H and all g_{i} belong to the subgroup

generated by s, t and H.

For every point x ∈ X_{(1)} set ∂_{x} = ∂_{x}^{y}, where y is the generic point of X and ∂^{x} = ∂_{z}^{x}, where z is the closed point of X. Thus,

((d_{X})^{2})_{z}^{y} = ∑_{x∈X_{(1)}} ∂^{x} ∘ ∂_{x} : K_{∗}(F ((s, t))) → K_{∗−2}(F).

To prove (i) let x_{s} and x_{t} be the points of X_{(1)} given by the ideals sR and tR, respectively. We have

∑_{x∈X_{(1)}} ∂^{x} ∘ ∂_{x}({s, t}) = ∂^{x_{s}}({t}) − ∂^{x_{t}}({s}) = 1 − 1 = 0.

To prove (ii) consider the field L = F ((s)) and the natural morphism

h : X′ = Spec(R[s^{−1}]) → Spec(L[t]) = 𝔸_{L}^{1}.

By the properties of marked polynomials, the map h identifies the set X′_{(0)} = X_{(1)} − {x_{s}} with the subset of the closed points of 𝔸_{L}^{1} given by irreducible marked polynomials. For every x ∈ X′ we write x̄ for the point h(x) ∈ 𝔸_{L}^{1}. Note that for x ∈ X′_{(0)} = X_{(1)} −{x_{s}}, the residue fields κ(x) and L(x̄) are canonically isomorphic. In particular, the field κ(x) can be viewed as a finite extension of L. By Fact 100.8(4), we have ∂^{x} = ∂ ∘ c_{κ(x)/L}, where ∂ : K_{∗}(L) → K_{∗−1}(F) is given by the canonical discrete valuation of L.

Let x ∈ X′_{(0)} = X_{(1)} − {x_{s}}. We write ∂x̄ for ∂ȳ ; x̄. Under the identification of κ(x) with L(x̄), we have ∂x̄ = ∂_{x} ∘ i where i : K_{∗}(L(t)) → K_{∗}(F ((s, t))) is the canonical homomorphism. Therefore,

∑_{x∈X_{(1)}} ∂^{x} ∘ ∂_{x} ∘ i = ∂^{x_{s}} ∘ ∂_{x_{s}} ∘ i + ∂ ∘ ∑_{x∈X′_{(0)}} c_{κ(x)/L} ∘ ∂_{x} ∘ i

= ∂^{x_{s}} ∘ ∂_{x_{s}} ∘ i + ∂ ∘ ∑_{x∈X′_{(0)}} c_{L(x̄)/L} ∘ ∂x̄.

Let α = {f, g_{1}, …, g_{n}} ∈ K_{n+1}(L(t)) with f and g_{i} as in (ii). Note that the divisors in 𝔸_{L}^{1} of the functions f and g_{i} are supported in the image of h. Hence ∂_{p}(α) = 0 for every closed point 𝔸_{L}^{1} not in the image of h. Moreover, for the point q of ℙ_{L}^{1} at infinity, f(q) = 1 and therefore, ∂_{q}(α) = 0. Hence, by Weil’s Reciprocity Law 49.10, applied to ℙ_{L}^{1},

∑_{x∈X′_{(0)}} c_{L(x̄)/L} ∘ ∂x̄(α) = ∑_{p∈ℙ_{L}^{1}} c_{L(p)/L} ∘ ∂_{p}(α) = 0.

Notice also that f(x_{s}) = 1, hence ∂_{x_{s}} ∘ i(α) = 0 and therefore,

(d_{X})^{2}({f, g_{1}, …, g_{n}}) = ∑_{x∈X_{(1)}} ∂^{x} ∘ ∂_{x} ∘ i(α) = 0.

Step 2: X = Spec(S), where S is a (noetherian) local complete 2-dimensional ring containing a field.

Let M ⊂ S be the maximal ideal. By Cohen’s theorem [140, Ch. VIII, Th.27], there is a subfield F ⊂ S such that the natural ring homomorphism F → S/M is an isomorphism.

Choose local parameters s, t ∈ M and consider the subring R = F [[s, t]] ⊂ S. Denote by P the maximal ideal of R. There is an integer r such that M^{r} ⊂ P S. We claim that the R-algebra S is finite. Indeed, note that

⋂_{n>0} P^{n}S ⊂ ⋂_{n>0} M^{n} = 0.

Since S/M^{r} is of finite length and there is a natural surjection S/M^{r} → S/P S, the ring S/P S is a finitely generated R/P -module. As the ring R is complete, S is a finitely generated R-module as claimed.

It follows from the claim that the natural morphism f : X → Y = Spec(R) is finite. By Proposition 49.9 and Step 1,

f_{∗} ∘ (d_{X})^{2} = (d_{Y})^{2} ∘ f_{∗} = 0.

The rings R and S have isomorphic residue fields, hence (d_{X})^{2} = 0.

Step 3: X = Spec(S) where S is a 2-dimensional (noetherian) local ring containing a field.

Let Ŝ be the completion of S. The natural morphism f : Y = Spec(Ŝ) → X is flat of relative dimension 0. By Proposition 49.23 and Step 2,

g^{∗} ∘ (d_{X})^{2} = (d_{Y})^{2} ∘ g^{∗} = 0.

The rings Ŝ and S have isomorphic residue fields, hence (d_{X})^{2} = 0.

Step 4: X is a scheme over a field.

Let x and x′ be two points of X such x′ is of codimension 2 in \overline{\{x\}}. We need to show that the (x, x′)-component of (d_{X})^{2} is trivial. We may assume that X = \overline{\{x\}}. The ring S = O_{X,x′} is local 2-dimensional. The natural morphism f : Y = Spec(S) → X is flat of constant relative dimension. By Proposition 49.23 and Step 3,

f^{∗} ∘ (d_{X})^{2} = (d_{Y})^{2} ∘ f^{∗} = 0.

The field κ(x′) and the residue field of S are isomorphic, therefore, the (x, x′)- component of (d_{X})^{2} is trivial. □

Definition 49.31. Let X be a scheme. The complex (C_{∗}(X), d_{X}) is called the Rost complex of X.

49.E. Boundary map. Let X be a scheme of finite type over a field and Z ⊂ X a closed subscheme. Set U = X \ Z. For every p ≥ 0, the set X_{(p)} is the disjoint union of Z_{(p)} and U_{(p)}, hence

C_{p}(X) = C_{p}(Z) ⊕ C_{p}(U).

Consider the closed embedding i : Z → X and the open immersion j : U → X. The sequence of complexes

0 → C_{∗}(Z) ⟶^{i_{∗}} C_{∗}(X) ⟶^{j^{∗}} C_{∗}(U) → 0

is exact. This sequence is not split in general as a sequence of complexes, but it splits canonically termwise. Let v : C_{∗}(U) → C_{∗}(X) and w : C_{∗}(X) → C_{∗}(Z) be the canonical inclusion and projection. Note that v and w do not commute with the differentials in general. We have j^{∗} ∘ v = id and w ∘ i_{∗} = id.

We define the boundary map

∂_{Z}^{U} : C_{p}(U) → C_{p−1}(Z)

by ∂_{Z}^{U} := w ∘ d_{X} ∘ v.

Example 49.32. Let X = 𝔸_{F}^{1} , Z = {0}, and U = 𝔾_{m} := 𝔸_{F}^{1} \ {0}. Then

∂_{Z}^{U}({t} · [U]) = [Z],

where t is the coordinate function on 𝔸_{F}^{1} .

Proposition 49.33. Let X be a scheme and Z ⊂ X a closed subscheme. Set U = X \ Z. Then d_{Z} ∘ ∂_{Z}^{U} = −∂_{Z}^{U} ∘ d_{U}.

Proof. By the definition of ∂ = ∂_{Z}^{U}, we have i_{∗} ∘ ∂ = d_{X} ∘ v − v ∘ d_{U}. Hence by Propositions 49.9 and 49.30,

i_{∗} ∘ d_{Z} ∘ ∂ = d_{X} ∘ i_{∗} ∘ ∂
= d_{X} ∘ (d_{X} ∘ v − v ∘ d_{U})
= − d_{X} ∘ v ∘ d_{U}
= (v ∘ d_{U} − d_{X} ∘ v) ∘ d_{U}
= − i_{∗} ∘ ∂ ∘ d_{U}.

Since i_{∗} is injective, we have d_{Z} ∘ ∂ = −∂ ∘ d_{U}. □

Proposition 49.34. Let a be an invertible function on X and let a′, a′′ be the restrictions of a on U and Z, respectively. Then

∂_{Z}^{U}(α · {a′}) = ∂_{Z}^{U}(α) · {a′′}  and  ∂_{Z}^{U}({a′} · α) = −{a′′} · ∂_{Z}^{U}(α)

for every α ∈ C_{∗}(U).

Proof. The homomorphisms v and w commute with the products. The statement follows from Proposition 49.5. □

Let

(49.35)  \begin{matrix}Z′ & ⟶^{i′} & X′ & ⟵^{j′} & U′ \\ ^{g}↓ & & ^{f}↓ & & ^{h}↓ \\ Z & ⟶^{i} & X & ⟵^{j} & U \end{matrix}

be a commutative diagram with i and i′ closed embeddings, j and j′ open embeddings and U = X \ Z, U′ = X′ \ Z′.

Proposition 49.36. Suppose that we have the diagram (49.35).

(1) If f, g and h are proper morphisms of schemes of finite type over a field, then the diagram

C_{p}(U′) ⟶^{∂_{Z′}^{U′}} C_{p−1}(Z′)
h_{∗}↓  ↓^{g_{∗}}
C_{p}(U) ⟶^{∂_{Z}^{U}} C_{p−1}(Z)

is commutative.

(2) Suppose that both squares in the diagram (49.35) are fiber squares. If f is flat of constant relative dimension d, then so are g and h and the diagram

C_{p}(U) ⟶^{∂_{Z}^{U}} C_{p−1}(Z)
h^{∗}↓  ↓^{g^{∗}}
C_{p+d}(U′) ⟶^{∂_{Z′}^{U′}} C_{p+d−1}(Z′)

is commutative.

Proof. (1) Consider the diagram

C_{p}(U′) ⟶^{v′} C_{p}(X′) ⟶^{d_{X′}} C_{p−1}(X′) ⟶^{w′} C_{p−1}(Z′)
h_{∗}↓  ^{f_{∗}}↓  ↓^{f_{∗}}  ↓^{g_{∗}}
C_{p}(U) ⟶^{v} C_{p}(X) ⟶^{d_{X}} C_{p−1}(X) ⟶^{w} C_{p−1}(Z).

The left and the right squares are commutative by the local nature of the definition of the push-forward homomorphisms. The middle square is commutative by Proposition 49.9.

The proof of (2) is similar — one uses Proposition 49.23. As both squares of the diagram are fiber squares, for any point z ∈ Z (respectively, u ∈ U), the fibers Z′_{z} and X′_{i(z)} (respectively, U′_{u} and X′_{j(u)}) are naturally isomorphic. □

Let Z_{1} and Z_{2} be closed subschemes of a scheme X. Set

T_{1} = Z_{1} \ Z_{2}, T_{2} = Z_{2} \ Z_{1}, U_{i} = X \ Z_{i}, U = U_{1} ∩ U_{2}, Z = Z_{1} ∩ Z_{2}.

We have the following fiber product diagram of open and closed embeddings:

Z ⟶ Z_{2} ⟵ T_{2}
↓  ↓  ↓
Z_{1} ⟶ X ⟵ U_{1}
↑  ↑  ↑
T_{1} ⟶ U_{2} ⟵ U.

Denote by ∂_{t}, ∂_{b}, ∂_{l}, ∂_{r} the boundary homomorphisms for the top, bottom, left and right triples of the diagram respectively.

Proposition 49.37. The morphism

∂_{l} ∘ ∂_{b} + ∂_{t} ∘ ∂_{r} : C_{∗}(U) → C_{∗−2}(Z)

is homotopic to zero.

Proof. The differential of C_{∗}(X) relative to the decomposition

C_{∗}(X) = C_{∗}(U) ⊕ C_{∗}(T_{1}) ⊕ C_{∗}(T_{2}) ⊕ C_{∗}(Z)

is given by the matrix

d_{X} = ( \begin{matrix}d_{U} & ∗ & ∗ & ∗ \\ ∂_{b} & ∗ & ∗ & ∗ \\ ∂_{r} & ∗ & ∗ & ∗ \\ h & ∂_{l} & ∂_{t} & d_{Z} \end{matrix} )

where h : C_{∗}(U) → C_{∗−1}(Z) is some morphism. The equality (d_{X})^{2} = 0 gives

h ∘ d_{U} + d_{Z} ∘ h + ∂_{l} ∘ ∂_{b} + ∂_{t} ∘ ∂_{r} = 0.

In other words, −h is a contracting homotopy for ∂_{l} ∘ ∂_{b} + ∂_{t} ∘ ∂_{r}. □

50. External products

From now on the word “scheme” means a separated scheme of finite type over a field.

Let X and Y be two schemes over F. We define the external product

C_{p}(X) × C_{q}(Y) → C_{p+q}(X × Y), (α, β) ↦ α × β

as follows: For a point v ∈ (X × Y)_{(p+q)}, we set (α × β)_{v} = 0 unless the point v projects to a point x in X_{(p)} and y in Y_{(q)}. In the latter case

(α × β)_{v} = l_{v} · r_{F (v)/F (x)}(α_{x}) · r_{F (v)/F (y)}(β_{y}),

where l_{v} is the length of the artinian local ring of v on Spec(F (x)) × Spec(F (y)).

The external product is graded symmetric with respect to X and Y. More precisely, if α ∈ C_{p,n}(X) and β ∈ C_{q,m}(Y), then

(50.1)  β × α = (−1)^{(p+n)(q+m)}(α × β).

For every point x ∈ X, we write Y_{x} for Y ×Spec(F (x)) and h_{x} for the canonical flat morphism Y_{x} → Y of relative dimension 0. Note that Y_{x} is a scheme over F (x), in particular, C_{∗}(Y_{x}) is a module over K_{∗}(F (x)). Denote by i_{x} : Y_{x} → X × Y the canonical morphism. Let α ∈ C_{p}(X) and β ∈ C_{q}(Y). Unfolding the definitions, we see that

α × β = ∑_{x∈X_{(p)}}(i_{x})_{∗}(α_{x} · (h_{x})^{∗}(β)).

Symmetrically, for every point y ∈ Y, we write X_{y} for X × Spec(F (y)) and k_{y} for the canonical flat morphism X_{y} → X of relative dimension 0. Note that X_{y} is a scheme over F (y), in particular, C_{∗}(X_{y}) is a module over K_{∗}(F (y)). Denote by j_{y} : X_{y} → X × Y the canonical morphism. Let α ∈ C_{p}(X) and β ∈ C_{q}(Y). Then

α × β = ∑_{y∈Y_{(q)}}(j_{y})_{∗}((k_{y})^{∗}(α) · β_{y}).

Proposition 50.2. For every α ∈ C_{∗}(X), β ∈ C_{∗}(Y), and γ ∈ C_{∗}(Z), we have

(α × β) × γ = α × (β × γ).

Proof. It is sufficient to show that for every point w ∈ (X × Y × Z)_{(p+q+r)} projecting to x ∈ X_{(p)}, y ∈ Y_{(q)} and z ∈ Z_{(r)} respectively, the w-components of both sides of the equality are equal to

r_{F (w)/F (x)}(α_{x}) · r_{F (w)/F (y)}(β_{y}) · r_{F (w)/F (z)}(γ_{z})

times the multiplicity that is the length of the local ring C of the point w on Spec(F (x)) × Spec(F (y)) × Spec(F (z)). Let v ∈ (X × Y)_{(p+q)} be the projection of w. The multiplicity of the v-component of α × β is equal to the length of the local ring B of the point v on Spec(F (x)) × Spec(F (y)). Clearly, C is flat over B. Let Q be the maximal ideal of B. The factor ring C/QC is the local ring of w on Spec(F (v)) × Spec(F (z)). Then the multiplicity of the w-component of the left hand side of the equality is equal to l(B) · l(C/QC). By Corollary 102.2, the latter number is equal to l(C). The multiplicity of the right hand side of the equality can be computed similarly. □

Proposition 50.3. For every α ∈ C_{p,n}(X) and β ∈ C_{q,m}(Y) we have

d_{X×Y} (α × β) = d_{X}(α) × β + (−1)^{p+n}α × d_{Y} (β).

Proof. We may assume that α ∈ K_{p+n}(F (x)) and β ∈ K_{q+m}(F (y)) for some points x ∈ X_{(p)} and y ∈ Y_{(q)}. For a point z ∈ (X × Y)_{(p+q−1)} the z-components of all three terms in the formula are trivial unless the projections of z to X and Y are specializations of x and y respectively. By dimension count, z projects either to x or to y.

Consider the first case. We have (d_{X}(α) × β)_{z} = 0. The point z belongs to the image of i_{x} and the morphism i_{x} factors as Y_{x} → \overline{\{x\}} × Y ↪ X × Y. The scheme Y_{x} is a localization of \overline{\{x\}} × Y. By Remark 49.7 and Proposition 49.9, the z-components of d_{X×Y} ∘ (i_{x})_{∗} and (i_{x})_{∗} ∘ d_{Y_{x}} are equal.

By Propositions 49.5 and 49.23, we have

[d_{X×Y} (α × β)]_{z} = [d_{X×Y} ∘ (i_{x})_{∗}(α · (h_{x})^{∗}(β))]_{z}
= [(i_{x})_{∗} ∘ d_{Y_{x}}(α · (h_{x})^{∗}(β))]_{z}
= (−1)^{p+n}[(i_{x})_{∗}(α · d_{Y_{x}} ∘ (h_{x})^{∗}(β))]_{z}
= (−1)^{p+n}[(i_{x})_{∗}(α · (h_{x})^{∗}(d_{Y} β))]_{z}
= (−1)^{p+n}[α × d_{Y} (β)]_{z}.

In the second case, symmetrically, we have (α × d_{Y} (β))_{z} = 0 and

d_{X×Y} (α × β)_{z} = (d_{X}(α) × β)_{z}.  □

Proposition 50.4. Let f : X → X′ and g : Y → Y′ be morphisms. Then for every α ∈ C_{p}(X) and β ∈ C_{q}(Y) we have

(f × g)_{∗}(α × β) = f_{∗}(α) × g_{∗}(β).

Proof. Clearly, it suffices to consider the case that f is the identity of X. Let x ∈ X_{(p)} and let i′_{x} : Y′_{x} → X × Y′, h′_{x} : Y′_{x} → Y′, and g_{x} : Y_{x} → Y′_{x} be canonical morphisms. We have

(1_{X} × g) ∘ i_{x} = i′_{x} ∘ g_{x} and g ∘ h_{x} = h′_{x} ∘ g_{x}.

By Propositions 49.8 and 49.20, we have

(1_{X} × g)_{∗}(α × β) = (1_{X} × g)_{∗} ∘ ∑(i_{x})_{∗}(α_{x} · (h_{x})^{∗}(β))
= ∑(i′_{x})_{∗} ∘ (g_{x})_{∗}(α_{x} · (h_{x})^{∗}(β))
= ∑(i′_{x})_{∗}(α_{x} · (g_{x})_{∗}(h_{x})^{∗}(β))
= ∑(i′_{x})_{∗}(α_{x} · (h′_{x})^{∗}g_{∗}(β))
= α × g_{∗}(β).  □

Proposition 50.5. Let f : X′ → X and g : Y′ → Y be flat morphisms. Then for every α ∈ C_{p}(X) and β ∈ C_{q}(Y) we have

(f × g)^{∗}(α × β) = f^{∗}(α) × g^{∗}(β).

Proof. Clearly, it suffices to consider the case that f is the identity of X. Let x ∈ X_{(p)} and let i′_{x} : Y′_{x} → X × Y′, h′_{x} : Y′_{x} → Y′ and g_{x} : Y′_{x} → Y_{x} be canonical morphisms. We have

(1_{X} × g) ∘ i′_{x} = i_{x} ∘ g_{x} and g ∘ h′_{x} = h_{x} ∘ g_{x}.

Note that the scheme Y_{x} is a localization of \overline{\{x\}} × Y. By Proposition 49.20 and Remark 49.22,

(1_{X} × g)^{∗} ∘ (i_{x})_{∗} = (i′_{x})_{∗} ∘ (g_{x})^{∗}.

By Propositions 49.16 and 49.18, we have

(1_{X} × g)^{∗}(α × β) = (1_{X} × g)^{∗} ∘ ∑(i_{x})_{∗}(α_{x} · (h_{x})^{∗}(β))
= ∑(i′_{x})_{∗} ∘ (g_{x})^{∗}(α_{x} · (h_{x})^{∗}(β))
= ∑(i′_{x})_{∗}(α_{x} · (g_{x})^{∗}(h_{x})^{∗}(β))
= ∑(i′_{x})_{∗}(α_{x} · (h′_{x})^{∗}g^{∗}(β))
= α × g^{∗}(β).  □

Corollary 50.6. Let f : X ×Y → X be the projection. Then for every α ∈ C_{∗}(X), we have f^{∗}(α) = α × [Y].

Proof. We apply Proposition 50.5 and Example 49.2 to f = 1_{X} × g, where g : Y → Spec(F) is the structure morphism. □

Proposition 50.7. Let X and Y be schemes over F. Let Z ⊂ X be a closed subscheme and U = X \ Z. Then for every α ∈ C_{p}(U) and β ∈ C_{q}(Y), we have

∂_{Z}^{U}(α) × β = ∂_{Z×Y}^{U×Y} (α × β).

Proof. We may assume that β ∈ K_{∗}(F (y)) for some y ∈ Y. By Propositions 49.36(1) and 50.4 we may also assume that Y = \overline{\{y\}}. For any scheme V denote by k^{V} : V_{y} → V and j^{V} : V_{y} → V × Y the canonical morphisms. Let v ∈ (Z × Y)_{(p+q−1)}. The v-component of both sides of the equality are trivial unless v belongs to the image of j^{Z}. By Remark 49.7, the v-component of j_{∗}^{Z} ∘ ∂_{Z_{y}}^{U_{y}} and ∂_{Z×Y}^{U×Y} ∘ j_{∗}^{U} are equal. It follows from Propositions 49.34 and 49.36(2) that

[∂_{Z}^{U}(α) × β]_{v} = [j_{∗}^{Z}((k^{Z})^{∗}(∂_{Z}^{U}α) · β)]_{v}
= [j_{∗}^{Z}(∂_{Z_{y}}^{U_{y}}(k^{U})^{∗}(α) · β)]_{v}
= [j_{∗}^{Z} ∘ ∂_{Z_{y}}^{U_{y}}((k^{U})^{∗}(α) · β)]_{v}
= [∂_{Z×Y}^{U×Y} ∘ j_{∗}^{U}((k^{U})^{∗}(α) · β)]_{v}
= [∂_{Z×Y}^{U×Y} (α × β)]_{v}.  □

Proposition 50.8. Let X and Y be two schemes and let a be an invertible regular function on X. Then for every α ∈ C_{p}(X) and β ∈ C_{q}(Y) we have

({a} · α) × β = {a′} · (α × β),

where a′ is the pull-back of a on X × Y.

Proof. Let ā be the pull-back of a on X_{y}. It follows from Propositions 49.8 and 49.16 that

({a} · α) × β = ∑(j_{y})_{∗}((k_{y})^{∗}({a}α) · β_{y})
= ∑(j_{y})_{∗}({ā}(k_{y})^{∗}(α) · β_{y})
= ∑{a′}(j_{y})_{∗}((k_{y})^{∗}(α) · β_{y})
= {a′} · (α × β).  □

51. Deformation homomorphisms

In this section we construct deformation homomorphisms . We shall use them later to define Gysin and pull-back homomorphisms. We follow Rost’s approach in [117] for the definition of deformation homomorphisms. (Deformation homomorphisms are called specialization homomorphism in [45].) Recall that we only consider separated schemes of finite type over a field.

Let f : Y → X be a closed embedding, D_{f} the deformation scheme and C_{f} the normal cone of f. Recall that D_{f} \ C_{f} is canonically isomorphic to 𝔾_{m} × X (cf. §104.E). We define the deformation homomorphism as the composition

σ_{f} : C_{∗}(X) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × X) ⟶^{{t}} C_{∗+1}(𝔾_{m} × X) ⟶^{∂} C_{∗}(C_{f})

where p : 𝔾_{m} × X → X is the projection, the coordinate t of 𝔾_{m} is considered as an invertible function on 𝔾_{m} × X and ∂ := ∂_{C_{f}}^{𝔾_{m}×X} is taken with respect to the open and closed subsets of the deformation scheme D_{f}.

Example 51.1. Let f = 1_{X} for a scheme X. Then D_{f} = 𝔸^{1} × X and C_{f} = X. We claim that σ_{f} is the identity. Indeed, it suffices to prove that the composition

C_{∗}(X) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × X) ⟶^{{t}} C_{∗+1}(𝔾_{m} × X) ⟶^{∂} C_{∗}(X)

is the identity. By Propositions 50.5, 50.7, 50.8, and Example 49.32, for every α ∈ C_{∗}(X), we have

∂({t} · p^{∗}(α)) = ∂({t} · ([𝔾_{m}] × α))
= ∂(({t} · [𝔾_{m}]) × α)
= ∂({t} · [𝔾_{m}]) × α
= [{0}] × α
= α.

The following statement is a consequence of Propositions 49.5, 49.23 and 49.33.

Proposition 51.2. Let f : Y → X be a closed embedding. Then σ_{f} ∘d_{X} = d_{C_{f}} ∘σ_{f}.

Consider the fiber product diagram

(51.3)  \begin{matrix}Y′ & ⟶^{f′} & X′ \\ ^{g}↓ & & ↓^{h} \\ Y & ⟶^{f} & X \end{matrix}

with f and f′ closed embeddings. Then we have the fiber product diagram (cf. §104.E)

(51.4)  \begin{aligned}C_{f′} & ⟶ & D_{f′} & ⟵ & 𝔾_{m} × X′ \\ ^{k}↓ & & ^{l}↓ & & ↓^{1×h} \\ C_{f} & ⟶ & D_{f} & ⟵ & 𝔾_{m} × X. \end{aligned}

Proposition 51.5. If the morphism h in diagram (51.3) is flat of relative dimension d then the morphism k in (51.4) is flat of relative dimension d and the diagram

C_{p}(X) ⟶^{σ_{f}} C_{p}(C_{f})
h^{∗}↓  ↓_{k^{∗}}
C_{p+d}(X′) ⟶^{σ_{f′}} C_{p+d}(C_{f′})

is commutative.

Proof. By Proposition 104.23, we have D_{f′} = D_{f} ×_{X}X′, hence the morphisms l and k in the diagram (51.4) are flat of relative dimension d. It follows from Propositions 49.16, 49.18, and 49.36(2) that the diagram

C_{∗}(X) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × X) ⟶^{{t}} C_{∗+1}(𝔾_{m} × X) ⟶^{∂} C_{∗}(C_{f})
h^{∗}↓  ^{(1×h)^{∗}}↓  ^{(1×h)^{∗}}↓  ↓_{k^{∗}}
C_{∗+d}(X′)⟶^{p^{∗}}C_{∗+d+1}(𝔾_{m} × X′)⟶^{{t}}C_{∗+d+1}(𝔾_{m} × X′)⟶^{∂}C_{∗+d}(C_{f′})

is commutative. □

Proposition 51.6. If the morphism h in (51.3) is a proper morphism, then the diagram

C_{∗}(X′) ⟶^{σ_{f′}} C_{∗}(C_{f′})
h_{∗}↓  ↓^{k_{∗}}
C_{∗}(X) ⟶^{σ_{f}} C_{∗}(C_{f})

is commutative.

Proof. The natural morphism D_{f′} → D_{f} ×_{X} X′ is a closed embedding by Proposition 104.23, hence the morphism l in the diagram (51.4) is proper. It follows from Propositions 49.8, 49.20, and 49.36(1) that the diagram

C_{∗}(X′) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × X′) ⟶^{{t}} C_{∗+1}(𝔾_{m} × X′) ⟶^{∂} C_{∗}(C_{f′})
h_{∗}↓  ^{(1×h)_{∗}}↓  ^{(1×h)_{∗}}↓  ↓^{k_{∗}}
C_{∗}(X) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × X) ⟶^{{t}} C_{∗+1}(𝔾_{m} × X) ⟶^{∂} C_{∗}(C_{f})

is commutative. □

Corollary 51.7. Let f : Y → X be a closed embedding. Then the composition σ_{f} ∘ f_{∗} coincides with the push-forward map C_{∗}(Y) → C_{∗}(C_{f}) for the zero section Y → C_{f}.

Proof. The statement follows from Proposition 51.6, applied to the fiber product square

\begin{matrix}Y & & Y \\ ∥ & & ^{f}↓ \\ Y & ⟶^{f} & X \end{matrix}

and Example 51.1. □

Lemma 51.8. Let f : X → 𝔸^{1} × W be a morphism. Suppose that the composition X → 𝔸^{1}×W → 𝔸^{1} and the restriction of f on both f^{−1}(𝔾_{m}×W) and f^{−1}({0}×W ) is flat. Then f is flat.

Proof. Let x ∈ X, y = f(x), and z ∈ 𝔸^{1} the projection of y. Set A = O_{𝔸^{1},z}, B = O_{𝔸^{1}×W,y}, and C = O_{X,x}. We need to show that C is flat over B. If z ≠ 0, this follows from the flatness of the restriction of f on f^{−1}(𝔾_{m} × W).

Suppose that z = 0. Let M be the maximal ideal of A. The rings B/MB and C/MC are the local rings of y on {0} × W and of x on f^{−1}({0} × W ) respectively. By assumption, C/MC is flat over B/MB and C is flat over A. It follows from [99, 20G] that C is flat over B. □

Lemma 51.9. Let f : U → V be a closed embedding and g : V → W a flat morphism. Suppose that the composition

q : C_{f} → U ⟶^{f} V ⟶^{g} W

is flat. Then σ_{f} ∘ g^{∗} = q^{∗}.

Proof. Consider the composition u : D_{f} → 𝔸^{1} × V ⟶^{1×g} 𝔸^{1} × W. The restriction of u on u^{−1}(𝔾_{m} × W) is isomorphic to 1 × g : 𝔾_{m} × V → 𝔾_{m} × W and is therefore flat. The restriction of u on u^{−1}(W × {0}) coincides with q and is flat by assumption. The projection D_{f} → 𝔸^{1} is also flat. It follows from Lemma 51.8 that the morphism u is flat.

Consider the fiber product diagram

C_{f} ⟶ D_{f} ⟵ 𝔾_{m} × V
q↓  ^{u}↓  ^{1×g}↓
W ⟶ 𝔸^{1} × W ⟵ 𝔾_{m} × W.

By Propositions 49.16, 49.18, and 49.36(2), the following diagram is commutative:

C_{∗}(W) ⟶^{p^{∗}} C_{∗+1}(𝔾_{m} × W) ⟶^{{t}} C_{∗+1}(𝔾_{m} × W) ⟶^{∂} C_{∗}(W)
g^{∗}↓  ^{1×g^{∗}}↓  ^{1×g^{∗}}↓  ↓^{q^{∗}}
C_{∗+d}(V) ⟶ C_{∗+d+1}(𝔾_{m} × V) ⟶^{{t}} C_{∗+d+1}(𝔾_{m} × V) ⟶^{∂} C_{∗+d}(C_{f}),

where d is the relative dimension of g. As the composition in the top row of the diagram is the identity by Example 51.1, the result follows. □

If f : Y → X is a regular closed embedding, we shall write N_{f} for the normal bundle C_{f}.

Let g : Z → Y and f : Y → X be regular closed embeddings. Then f ∘ g : Z → X is also a regular closed embedding by Proposition 104.15. The normal bundles of the regular closed embeddings i : N_{f}|_{Z} → N_{f} and j : N_{g} → N_{f∘g} are canonically isomorphic; we denote them by N (cf. §104.E).

Lemma 51.10. In the setup above, the morphisms of complexes σ_{i} ∘ σ_{f} and σ_{j} ∘ σ_{f∘g} : C_{∗}(X) → C_{∗}(N) are homotopic.

Proof. Let D = D_{f,g} be the double deformation scheme (cf. §104.F). We have the following fiber product diagram of open and closed embeddings:

N  ⟶  D_{i}  ⟵ N_{f} × 𝔾_{m}
↓  ↓  ↓
D_{j}  ⟶  D  ⟵ D_{f} × 𝔾_{m}
↑  ↑  ↑
𝔾_{m} × N_{f∘g} ⟶ 𝔾_{m} × D_{fg} ⟵ 𝔾_{m} × X × 𝔾_{m}.

We shall use the notation ∂_{t}, ∂_{b}, ∂_{l}, ∂_{r} for the boundary morphisms as in §49.E. For every scheme V, denote by p_{V} either of the projections V × 𝔾_{m} → V or 𝔾_{m} × V → V. Write p for the projection 𝔾_{m} × X × 𝔾_{m} → X.

By Proposition 49.34 and 51.5, we have

σ_{i} ∘ σ_{f} = ∂_{t} ∘ {s} ∘ p_{N_{f}}^{∗} ∘ σ_{f}
= ∂_{t} ∘ {s} ∘ σ_{f×𝔾_{m}} ∘ p_{X}^{∗}
= ∂_{t} ∘ {s} ∘ ∂_{r} ∘ {t} ∘ p^{∗}
= − ∂_{t} ∘ ∂_{r} ∘ {s, t} ∘ p^{∗},

and similarly,

σ_{j} ∘ σ_{fg} = ∂_{l} ∘ {t} ∘ p_{N_{f∘g}}^{∗} ∘ σ_{fg}
= ∂_{l} ∘ {t} ∘ σ_{𝔾_{m}×f∘g} ∘ p_{X}^{∗}
= ∂_{l} ∘ {t} ∘ ∂_{b} ∘ {s} ∘ p^{∗}
= − ∂_{l} ∘ ∂_{b} ∘ {t, s} ∘ p^{∗}.

As {s, t} = −{t, s} (cf. §49.B) and the compositions ∂_{t} ∘ ∂_{r} and −∂_{l} ∘ ∂_{b} are homotopic by Proposition 49.37, the result follows. □

52. K-homology groups

Let X be a separated scheme of finite type over a field F. The complex C_{∗}(X) is the coproduct of complexes C_{∗q}(X) over all q ∈ ℤ. Denote the pth homology group of the complex C_{∗q}(X) by A_{p}(X, K_{q}) and call it the K-homology group of X. In other words, A_{p}(X, K_{q}) is the homology group of the complex

∐_{dim x=p+1} K_{p+q+1}(F (x)) ⟶^{d_{X}} ∐_{dim x=p} K_{p+q}(F (x)) ⟶^{d_{X}} ∐_{dim x=p−1} K_{p+q−1}(F (x)).

It follows from the definition that A_{p}(X, K_{q}) = 0 if p+q < 0, p < 0, or p > dim X.

The group A_{p}(X, K_{−p}) is a factor group of ∐_{dim x=p} K_{0}(F (x)). If Z ⊂ X is a closed subscheme, the coset of the cycle [Z] of Z in A_{p}(X, K_{−p}) (cf. Example 49.2) will be also denoted by [Z].

52. K-HOMOLOGY GROUPS 239

If X is the disjoint union of two schemes X_{1} and X_{2}, then by (49.4), we have

A_{p}(X, K_{q}) = A_{p}(X_{1}, K_{q}) ⊕ A_{p}(X_{2}, K_{q}).

Example 52.1. We have

A_{p}(Spec(F), K_{q}) = \{ \begin{matrix}K_{q}(F) & if p = 0, \\ 0 & otherwise. \end{matrix}

It follows from Fact 100.5 that

A_{p}(𝔸_{F}^{1} , K_{q}) = \{ \begin{matrix}K_{q+1}(F) & if p = 1, \\ 0 & otherwise. \end{matrix}

52.A. Push-forward homomorphisms. If f : X → Y is a proper morphism of schemes, the push-forward homomorphism f_{∗} : C_{∗q}(X) → C_{∗q}(Y) is a morphism of complexes by Proposition 49.9. This induces the push-forward homomorphism of K-homology groups

(52.2)  f_{∗} : A_{p}(X, K_{q}) → A_{p}(Y, K_{q}).

Thus, the assignment X ↦ A_{∗}(X, K_{∗}) gives rise to a functor from the category of schemes and proper morphisms to the category of bigraded abelian groups and bigraded homomorphisms.

Example 52.3. Let f : X → Y be a closed embedding such that f is a bijection on points. It follows from Example 49.6 that the push-forward homomorphism f_{∗} in (52.2) is an isomorphism.

52.B. Pull-back homomorphism. If g : Y → X is a flat morphism of relative dimension d, the pull-back homomorphism g^{∗} : C_{∗q}(X) → C_{∗+d,q−d}(Y) is a morphism of complexes by Proposition 49.23. This induces the pull-back homomorphism of the K-homology groups

g^{∗} : A_{p}(X, K_{q}) → A_{p+d}(Y, K_{q−d}).

The assignment X ↦ A_{∗}(X, K_{∗}) gives rise to a contravariant functor from the category of schemes and flat morphisms to the category of abelian groups.

Example 52.4. If X is a variety of dimension d over F, then the flat structure morphism p : X → Spec(F) of relative dimension d induces a natural pull-back homomorphism

p^{∗} : K_{q}F = A_{0}(Spec(F), K_{q}) → A_{d}(X, K_{q−d})

giving A_{∗}(X, K_{∗}) a structure of a K_{∗}(F)-module.

Example 52.5. It follows from Example 52.1 that the pull-back homomorphism

f^{∗} : A_{p}(Spec(F), K_{q}) → A_{p+1}(𝔸_{F}^{1} , K_{q−1})

given by the flat structure morphism f : 𝔸_{F}^{1} → Spec(F) is an isomorphism.

52.C. Product. Let X and Y be two schemes over F. It follows from Proposition 50.3 that there is a well-defined pairing

A_{p}(X, K_{n}) ⊗ A_{q}(Y, K_{m}) → A_{p+q}(X × Y, K_{n+m})

taking the classes of cycles α and β to the class of the external product α × β (cf. §50).

52.D. Localization. Let X be a scheme and Z ⊂ X a closed subscheme. Set U := X \ Z and consider the closed embedding i : Z → X and the open immersion j : U → X. The exact sequence of complexes

0 → C_{∗}(Z) ⟶^{i_{∗}} C_{∗}(X) ⟶^{j^{∗}} C_{∗}(U) → 0

induces the long localization exact sequence of K-homology groups

(52.6) … ⟶^{δ} A_{p}(Z, K_{q}) ⟶^{i_{∗}} A_{p}(X, K_{q}) ⟶^{j^{∗}} A_{p}(U, K_{q}) ⟶^{δ} A_{p−1}(Z, K_{q}) ⟶^{i_{∗}} …

The map δ is called the connecting homomorphism. It is induced by the boundary map of complexes ∂_{Z}^{U} : C_{∗}(U) → C_{∗−1}(Z) (cf. Proposition 49.33).

52.E. Deformation. Let f : Y → X be a closed embedding. It follows from Proposition 51.2 that the deformation homomorphism σ_{f} of complexes induces the deformation homomorphism of homology groups

σ_{f} : A_{p}(X, K_{q}) → A_{p}(C_{f}, K_{q}),

where C_{f} is the normal cone of f.

Proposition 52.7. Let Z be a closed equidimensional subscheme of a scheme X and g : f^{−1}(Z) → Z the restriction of f. Then σ_{f}([Z]) = h_{∗}([C_{g}]), where h : C_{g} → C_{f} is the closed embedding of cones.

Proof. Let i : Z → X be the closed embedding and q : Z → Spec(F), r : C_{f} → Spec(F) the structure morphisms. Consider the diagram

A_{0}(Spec(F), K_{0}) ⟶^{q^{∗}} A_{d}(Z, K_{−d}) ⟶^{i_{∗}} A_{d}(X, K_{−d})
∥  ^{σ_{g}} ↓  ^{σ_{f}} ↓
A_{0}(Spec(F), K_{0}) ⟶^{r^{∗}} A_{d}(C_{g}, K_{−d}) ⟶^{h_{∗}} A_{d}(C_{f}, K_{−d}),

where d = dim Z. The left square is commutative by Lemma 51.9 and the right one by Proposition 51.6. Consequently, σ_{f}([Z]) = σ_{f} ∘ i_{∗} ∘ q^{∗}(1) = h_{∗} ∘ r^{∗}(1) = h_{∗}([C_{g}]). □

52.F. Continuity. Let X be a variety of dimension n and f : Y → X a dominant morphism. Let x denote the generic point of X and Y_{x} the generic fiber of f. For every nonempty open subscheme U ⊂ X, the natural flat morphism g_{U} : Y_{x} → f^{−1}(U) is of relative dimension −n. Hence we have the pull-back homomorphism

(52.8)  g_{U}^{∗} : C_{∗}(f^{−1}(U)) → C_{∗−n}(Y_{x}).

The following proposition is a straightforward consequence of the definition of the complexes C_{∗}.

Proposition 52.9. The pull-back homomorphism g_{U}^{∗} of (52.8) induces isomorphisms

colim C_{p}(f^{−1}(U)) →^{∼} C_{p−n}(Y_{x}) and colim A_{p}(f^{−1}(U), K_{q}) →^{∼} A_{p−n}(Y_{x}, K_{q+n})

for all p and q, where the colimits are taken over all nonempty open subschemes U of X.

52.G. Homotopy invariance. Let g : Y → X be a morphism of schemes over F. Recall that for every x ∈ X, we denote by Y_{x} the fiber scheme g^{−1}(x) = Y ×_{X} Spec(F (x)) over the field F (x).

Proposition 52.10. Let g : Y → X be a flat morphism of relative dimension d. Suppose that for every x ∈ X, the pull-back homomorphism

A_{p}(Spec(F (x)), K_{q}) → A_{p+d}(Y_{x}, K_{q−d})

is an isomorphism for every p and q. Then the pull-back homomorphism

g^{∗} : A_{p}(X, K_{q}) → A_{p+d}(Y, K_{q−d})

is an isomorphism for every p and q.

Proof. Step 1: X is a variety.

We induct on n = dim X. The case n = 0 is obvious. In general, let U ⊂ X be a nonempty open subset and Z = X \U with the structure of a reduced scheme. Set V = g^{−1}(U) and T = g^{−1}(Z). We have closed embeddings i : Z → X, k : T → Y and open immersions j : U → X, l : V → Y. By induction, the pull-back homomorphism (g|_{T})^{∗} in the diagram

A_{p+1}(U, K_{q}) ⟶^{δ} A_{p}(Z, K_{q}) ⟶^{i_{∗}} A_{p}(X, K_{q})
(g|_{V} )^{∗}↓  ^{(g|_{T})^{∗}}↓  ^{g^{∗}}↓
A_{p+d+1}(V, K_{q−d}) ⟶^{δ} A_{p+d}(T, K_{q−d}) ⟶^{k_{∗}} A_{p+d}(Y, K_{q−d})
A_{p}(X, K_{q}) ⟶^{j^{∗}} A_{p}(U, K_{q}) ⟶^{δ} A_{p−1}(Z, K_{q})
g^{∗}↓  ^{(g|_{V} )^{∗}}↓  ^{(g|_{T})^{∗}}↓
A_{p+d}(Y, K_{q−d}) ⟶^{l^{∗}} A_{p+d}(V, K_{q−d}) ⟶^{δ} A_{p+d−1}(T, K_{q−d})

is an isomorphism. The diagram is commutative by Propositions 49.18, 49.20, and 49.36(2).

Let x ∈ X be the generic point. By Proposition 52.9, the colimit of the homomorphisms

(g|_{V})^{∗} : A_{p}(U, K_{q}) → A_{p+d}(V, K_{q−d})

over all nonempty open subschemes U of X is isomorphic to the pull-back homomorphism

A_{p−n}(Spec(F (x)), K_{q+n}) → A_{p−n+d}(Y_{x}, K_{q+n−d}).

By assumption, it is an isomorphism. Taking the colimits of all terms of the diagram, we conclude by the 5-lemma that g^{∗} is an isomorphism.

Step 2: X is reduced.

We induct on the number m of irreducible components of X. The case m = 1 is Step 1. Let Z be a (reduced) irreducible component of X and let U = X \ Z. Consider the commutative diagram as in Step 1. By Step 1, we have (g|_{T})^{∗} is an isomorphism. The pull-back (g|_{V})^{∗} is also an isomorphism by the induction hypothesis. By the 5-lemma, g^{∗} is an isomorphism.

Step 3: X is an arbitrary scheme.

Let X′ be the reduced scheme X_{red}. Consider the fiber product diagram

Y′ ⟶^{g′} X′
f↓  ↓^{h}
Y ⟶^{g} X,

with f and h closed embeddings. By Proposition 49.20, we have g^{∗}∘h_{∗} = f_{∗}∘g′^{∗}. It follows from Example 52.3 that the maps f_{∗} and h_{∗} are isomorphisms. Finally, g′^{∗} is an isomorphism by Step 2, hence we conclude that g^{∗} is also an isomorphism. □

Corollary 52.11. The pull-back homomorphism

g^{∗} : A_{p}(X, K_{q}) → A_{p+d}(X × 𝔸_{F}^{d} , K_{q−d})

induced by the projection g : X × 𝔸_{F}^{d} → X is an isomorphism. In particular,

A_{p}(𝔸^{d}, K_{q}) = \{ \begin{matrix}K_{q+d}(F) & if p = d, \\ 0 & otherwise. \end{matrix}

Proof. Example 52.5 and Proposition 52.10 yield the statement in the case d = 1. The general case follows by induction. □

A morphism g : Y → X is called an affine bundle of rank d if g is flat and the fiber of g over any point x ∈ X is isomorphic to the affine space 𝔸_{F (x)}^{d}. For example, a vector bundle of rank d is an affine bundle of rank d.

The following statement is a useful criterion establish a morphism is an affine bundle. We shall use it repeatedly in the sequel.

Lemma 52.12. A morphism f : Y → X over F is an affine bundle of rank d if for any local commutative F -algebra R and any morphism Spec(R) → X over F, the fiber product Y ×_{X} Spec(R) is isomorphic to 𝔸_{R}^{d} over R.

Proof. Applying the condition to the local ring R = O_{X,x} for each x ∈ X, we see that f is flat and the fiber of f over x is the affine space 𝔸_{F (x)}^{d}. □

The following theorem implies that affine spaces are essentially negligible for K-homology computation.

Theorem 52.13 (Homotopy Invariance). Let g : Y → X be an affine bundle of rank d. Then the pull-back homomorphism

g^{∗} : A_{p}(X, K_{q}) → A_{p+d}(Y, K_{q−d})

is an isomorphism for every p and q.

Proof. For every x ∈ X, we have Y_{x} ≃ 𝔸_{F (x)}^{d}. Applying Corollary 52.11 to X = Spec(F (x)), we see that the pull-back homomorphism

A_{p}(Spec(F (x)), K_{q}) → A_{p+d}(Y_{x}, K_{q−d})

is an isomorphism for every p and q. By Proposition 52.10, the map g^{∗} is an isomorphism. □

Corollary 52.14. Let f : E → X be a vector bundle of rank d. Then the pull-back homomorphism

f^{∗} : A_{p}(X, K_{∗}) → A_{p+d}(E, K_{∗−d})

is an isomorphism for every p.

53. Euler classes and projective bundle theorem

In this section, we compute the K-homology for projective spaces and more generally for projective bundles.

53.A. Euler class. Let p : E → X be a vector bundle of rank r. Let s : X → E denote the zero section. Note that p is a flat morphism of relative dimension r and s is a closed embedding. By Corollary 52.14, the pull-back homomorphism p^{∗} is an isomorphism. We call the composition

e(E) = (p^{∗})^{−1} ∘ s_{∗} : A_{∗}(X, K_{∗}) → A_{∗−r}(X, K_{∗+r})

the Euler class of E. Note that isomorphic vector bundles over X have equal Euler classes.

Proposition 53.1. Let 0 → E′ ⟶^{f} E ⟶^{g} E′′ → 0 be an exact sequence of vector bundles over X. Then e(E) = e(E′′) ∘ e(E′).

Proof. Consider the fiber product diagram

E′ ⟶^{f} E
p′↓  ^{g}↓
X ⟶^{s′′} E′′.

By Proposition 49.20, we have g^{∗} ∘ s′′_{∗} = f_{∗} ∘ p′^{∗}, hence

e(E′′) ∘ e(E′) =(p′′^{∗})^{−1} ∘ s′′_{∗} ∘ (p′^{∗})^{−1} ∘ s′_{∗}
=(p′′^{∗})^{−1} ∘ g^{∗}^{−1} ∘ f_{∗} ∘ s′_{∗}
=(p′′ ∘ g)^{∗}^{−1} ∘ (f ∘ s′)_{∗}
=p^{∗}^{−1} ∘ s_{∗}
=e(E).  □

Corollary 53.2. The Euler classes of any two vector bundles E and E′ over X commute: e(E′) ∘ e(E) = e(E) ∘ e(E′).

Proof. By Proposition 53.1, we have

e(E′) ∘ e(E) = e(E′ ⊕ E) = e(E ⊕ E′) = e(E) ∘ e(E′).  □

Proposition 53.3. Let f : Y → X be a morphism and E a vector bundle over X. Then the pull-back E′ = f^{∗}(E) is a vector bundle over Y and

(1) If f is proper, then e(E) ∘ f_{∗} = f_{∗} ∘ e(E′).

(2) If f is flat, then f^{∗} ∘ e(E) = e(E′) ∘ f^{∗}.

Proof. We have two fiber product diagrams

E′ ⟶^{g} E  Y ⟶^{f} X
q↓  ↓^{p}  and  ^{j}↓  ↓^{i}
Y ⟶^{f} X  E′ ⟶^{g} E

where p and q are the natural morphisms and i and j are the zero sections.

(1): By Proposition 49.20, we have p^{∗} ∘ f_{∗} = g_{∗} ∘ q^{∗}. Hence
e(E) ∘ f_{∗} = (p^{∗})^{−1} ∘ i_{∗} ∘ f_{∗}
= (p^{∗})^{−1} ∘ g_{∗} ∘ j_{∗}
= f_{∗} ∘ (q^{∗})^{−1} ∘ j_{∗}
= f_{∗} ∘ e(E′).
(2): Again by Proposition 49.20, we have g^{∗} ∘ i_{∗} = j_{∗} ∘ f^{∗}. Hence
f^{∗} ∘ e(E) = f^{∗} ∘ (p^{∗})^{−1} ∘ i_{∗}
= (q^{∗})^{−1} ∘ g^{∗} ∘ i_{∗}
= (q^{∗})^{−1} ∘ j_{∗} ∘ f^{∗}
= e(E′) ∘ f^{∗}.  □

Proposition 53.4. Let p : E → X and p′ : E′ → X′ be vector bundles. Then

e(E × E′)(α × α′) = e(E)(α) × e(E′)(α′)

for every α ∈ A_{∗}(X, K_{∗}) and α′ ∈ A_{∗}(X′, K_{∗}).

Proof. Let s : X → E and s′ : X′ → E′ be zero sections. It follows from Propositions 50.4 and 50.5 that

e(E × E′)(α × α′) = (p × p′)^{∗}^{−1} ∘ (s × s′)_{∗}(α × α′)
= (p^{∗}^{−1} × p′^{∗}^{−1}) ∘ (s_{∗} × s′_{∗})(α × α′)
= (p^{∗}^{−1} ∘ s_{∗}(α)) × (p′^{∗}^{−1} ∘ s′_{∗}(α′))
= e(E)(α) × e(E′)(α′).  □

Proposition 53.5. The Euler class e(𝟙) is trivial.

Proof. It suffices to proof that the push-forward homomorphism s_{∗} for the zero section s : X → 𝔸^{1} × X is trivial. Let t be the coordinate function on 𝔸^{1}. We view {t} as an element of C_{1}(𝔸^{1}) = K_{1}(F (𝔸^{1})). Clearly, d_{𝔸^{1}}({t}) = div(t) = [0]. It follows from Proposition 50.3 that for every α ∈ A^{∗}(X, K_{∗}), one has

s_{∗}(α) = [0] × α = d_{𝔸^{1}}({t}) × α = d_{𝔸^{1}×X}({t} × α) = 0

in A^{∗}(𝔸^{1} × X, K_{∗}) □

53.B. K-homology of projective spaces. Let V be a vector space of dimension d + 1 over F, X the projective space ℙ_{F} (V) and V_{p} a subspace of V of dimension p + 1 for p = 0, …, d. We view ℙ(V_{p}) as a subvariety of X. Let x_{p} ∈ X be the generic point of ℙ(V_{p}). Consider the generator 1_{p} of K_{0}(F (x_{p})) = ℤ viewed as a subgroup of C_{p,−p}(X). We claim that the class l_{p} of the generator 1_{p} in A_{p}(X, K_{−p}) does not depend on the choice of V_{p}.

The statement is trivial if p = d. Suppose p < d and let V′_{p} be another subspace of dimension p + 1. We may assume that V_{p} and V′_{p} are subspaces of a subspace W ⊂ V of dimension p+2. Let h and h′ be linear forms on W satisfying Ker(h) = V_{p} and Ker(h′) = V′_{p}. View the ratio f = h/h′ as a rational function on ℙ_{F} (W), so div(f) = 1_{p} − 1_{p′}. By definition of the K-homology group A_{p}(X, K_{−p}), the classes l_{p} and l_{p′} of 1_{p} and 1_{p′} respectively in A_{p}(X, K_{−p}) coincide.

Let X be a scheme over F and set ℙ_{X}^{d} = ℙ_{F}^{d} × X. For every i = 0, …, d consider the external product homomorphism

A_{∗−i}(X, K_{∗+i}) → A_{∗}(ℙ_{X}^{d}, K_{∗}), α ↦ l_{i} × α.

The following proposition computes the K-homology of the projective space ℙ_{X}^{d}.

Proposition 53.6. For any scheme X, the homomorphism

∐_{i=0}^{d} A_{∗−i}(X, K_{∗+i}) → A_{∗}(ℙ_{X}^{d}, K_{∗})

taking ∑ α_{i} to ∑ l_{i} × α_{i} is an isomorphism.

Proof. We induct on d. The case d = 0 is obvious since ℙ_{X}^{d} = X. If d > 0 we view ℙ_{X}^{d−1} as a closed subscheme of ℙ_{X}^{d} with open complement 𝔸_{X}^{d}. Consider the closed and open embeddings f : ℙ_{X}^{d−1} → ℙ_{X}^{d} and g : 𝔸_{X}^{d} → ℙ_{X}^{d}. In the diagram

0 ⟶ ∐_{i=0}^{d−1} A_{∗−i}(X, K_{∗+i}) ⟶ ∐_{i=0}^{d} A_{∗−i}(X, K_{∗+i})
↓  ↓
… ⟶^{δ}  A_{∗}(ℙ_{X}^{d−1}, K_{∗})  ⟶^{f_{∗}}  A_{∗}(ℙ_{X}^{d}, K_{∗})
∐_{i=0}^{d} A_{∗−i}(X, K_{∗+i}) ⟶ A_{∗−d}(X, K_{∗+d}) ⟶ 0
↓  _{h^{∗}}↓
A_{∗}(ℙ_{X}^{d}, K_{∗})  ⟶^{g^{∗}} A_{∗}(𝔸_{X}^{d}, K_{∗}) ⟶^{δ} …

the bottom row is the localization exact sequence and h : 𝔸_{X}^{d} → X is the canonical morphism. The left square is commutative by Proposition 50.4 and the right square by Proposition 50.5.

Let q : ℙ_{X}^{d} → X be the projection. Since h = q ∘ g, we have h^{∗} = g^{∗} ∘ q^{∗}. By Corollary 52.11, we have h^{∗} is an isomorphism, hence g^{∗} is surjective. Therefore, all connecting homomorphisms δ in the bottom localization exact sequence are trivial. It follows that the map f_{∗} is injective, i.e., the bottom sequence of the two maps f_{∗} and g^{∗} is short exact. By the induction hypothesis, the left vertical homomorphism is an isomorphism. By the 5-lemma so is the middle one. □

Corollary 53.7. A_{p}(ℙ_{F}^{d} , K_{q}) = \begin{cases}K_{p+q}(F) · l_{p} & if 0 ≤ p ≤ d, \\ 0 & otherwise. \end{cases}

Example 53.8. Let L be the canonical line bundle over X = ℙ_{F}^{d}. We claim that e(L)(l_{p}) = l_{p−1} for every p = 1, …, d. First consider the case p = d. By §104.C, we have L = ℙ^{d+1} \ {0}, where 0 = [0 : … 0 : 1] and the morphism f : L → X takes [S_{0} : ⋯: S_{n} : S_{n+1}] to [S_{0} : ⋯: S_{n}]. The image Z of the zero section s : X → L is given by S_{n+1} = 0. Let H ⊂ X be the hyperplane given by S_{0} = 0. We have div(S_{n+1}/S_{0}) = [Z] − [f^{−1}(H)] and therefore, in A_{d−1}(X, K_{1−d}):

e(L)(l_{d}) = (f^{∗})^{−1}(s_{∗}([X])) = (f^{∗})^{−1}([Z]) = (f^{∗})^{−1}([f^{−1}(H)]) = [H] = l_{d−1}.

In the general case consider a linear closed embedding g : ℙ_{F}^{p} → ℙ_{F}^{d}. The pullback L′ := g^{∗}(L) is the canonical bundle over ℙ_{F}^{p}. By the first part of the proof and Proposition 53.3(1),

e(L)(l_{p}) = e(L)(g_{∗}(l_{p})) = g_{∗}(e(L′)(l_{p})) = g_{∗}(l_{p−1}) = l_{p−1}.

Example 53.9. Let L′ be the tautological line bundle over X = ℙ_{F}^{d}. Similar to Example 53.8, we get e(L′)(l_{p}) = −l_{p−1} for every p ∈ [1, d].

53.C. Projective bundle theorem. Let E → X be a vector bundle of rank r > 0. Consider the associated projective bundle morphism q : ℙ(E) → X. Note that q is a flat morphism of relative dimension r − 1. Let L → ℙ(E) be either the canonical or the tautological line bundle and e the Euler class of L.

Theorem 53.10 (Projective Bundle Theorem). Let E → X be a vector bundle of rank r > 0. Then the homomorphism

φ(E) := ∐_{i=1}^{r} e^{r−i} ∘ q^{∗} : ∐_{i=1}^{r} A_{∗−i+1}(X, K_{∗+i−1}) → A_{∗}(ℙ(E), K_{∗})

is an isomorphism. In other words, every α ∈ A_{∗}(ℙ(E), K_{∗}) can be written in the form

α = ∑_{i=1}^{r} e^{r−i}(q^{∗}(α_{i}))

for uniquely determined elements α_{i} ∈ A_{∗−i+1}(X, K_{∗+i−1}).

Proof. Suppose that L is the canonical line bundle; the case of the tautological bundle is treated similarly. If E is a trivial vector bundle, we have ℙ(E) = X×ℙ_{F}^{r−1}. Let L′ be the canonical line bundle over ℙ_{F}^{r−1}. It follows from Example 53.8 that

e(L)^{r−i}(q^{∗}(α)) = e(L)^{r−i}(l_{r−1} × α)
= e(L′)^{r−i}(l_{r−1}) × α
= l_{i−1} × α.

Hence the map φ(E) coincides with the one in Proposition 53.6, consequently is an isomorphism.

In general, we induct on d = dim X. If d = 0, the vector bundle is trivial. If d > 0 choose an open subscheme U ⊂ X such that dimension of Z = X \ U is less than d and the vector bundle E|_{U} is trivial. In the diagram

… ⟶ ∐_{i=1}^{r} A_{∗−i+1}(Z, K_{∗+i−1}) ⟶ ∐_{i=1}^{r} A_{∗−i+1}(X, K_{∗+i−1})
φ(E|_{Z})↓  ^{φ(E)}↓
… ⟶  A_{∗}(ℙ(E|_{Z}), K_{∗})  ⟶  A_{∗}(ℙ(E), K_{∗})

∐_{i=1}^{r} A_{∗−i+1}(X, K_{∗+i−1}) ⟶ ∐_{i=1}^{r} A_{∗−i+1}(U, K_{∗+i−1}) ⟶ …
φ(E)↓  ^{φ(E|_{U})}↓
A_{∗}(ℙ(E), K_{∗})  ⟶  A_{∗}(ℙ(E|_{U}), K_{∗})  ⟶ …

with rows the localization long exact sequences, the homomorphisms φ(E|_{Z}) are isomorphisms by the induction hypothesis and φ(E|_{U}) are isomorphisms as E|_{U} is trivial. The diagram is commutative by Proposition 53.3. The statement follows by the 5-lemma. □

Remark 53.11. It follows from Propositions 49.18, 49.20, and 53.3 that the isomorphisms φ(E) are natural with respect to push-forward homomorphisms for proper morphisms of the base schemes and with respect to pull-back homomorphisms for flat morphisms.

Corollary 53.12. The pull-back homomorphism

q^{∗} : A_{∗−r+1}(X, K_{∗+r−1}) → A_{∗}(ℙ(E), K_{∗})

is a split injection.

Proposition 53.13 (Splitting Principle). Let E → X be a vector bundle. Then there is a flat morphism f : Y → X of constant relative dimension, say d, such that:

(1) The pull-back homomorphism f^{∗} : A_{∗}(X, K_{∗}) → A_{∗+d}(Y, K_{∗−d}) is injective.

(2) The vector bundle f^{∗}(E) has a filtration by subbundles with quotients line bundles.

Proof. We induct on the rank r of E. Consider the projective bundle q : ℙ(E) → X. The pull-back homomorphism q^{∗} is injective by Corollary 53.12. The tautological line bundle L over ℙ(E) is a sub-bundle of the vector bundle q^{∗}(E). Applying the induction hypothesis to the factor bundle q^{∗}(E)/L over ℙ(E), we find a flat morphism g : Y → ℙ(E) of constant relative dimension satisfying the conditions (1) and (2). The composition f = q ∘ g works. □

To prove various relations between K-homology classes, the splitting principle allows us to assume that all the vector bundles involved have filtration by subbundles with line factors.

54. Chern classes

In this section we construct Chern classes of vector bundles as operations on K-homology.

Let E → X be a vector bundle of rank r > 0 and q : ℙ(E) → X the associated projective bundle. By Theorem 53.10, for every α ∈ A_{∗}(X, K_{∗}) there exist unique α_{i} ∈ A_{∗−i}(X, K_{∗+i}), i = 0, …, r such that

−e^{r}(q^{∗}(α)) = ∑_{i=1}^{r}(−1)^{i}e^{r−i}(q^{∗}(α_{i})),

where e is the Euler class of the tautological line bundle L over ℙ(E), i.e.,

(54.1)  ∑_{i=0}^{r}(−1)^{i}e^{r−i}(q^{∗}(α_{i})) = 0,

where α_{0} = α. Thus we have obtained group homomorphisms

(54.2)  c_{i}(E) : A_{∗}(X, K_{∗}) → A_{∗−i}(X, K_{∗+i}), α ↦ α_{i} = c_{i}(E)(α)

for every i = 0, …, r. These are called the Chern classes of E. By definition, c_{0}(E) is the identity. We also set c_{i} = 0 if i > r or i < 0 and define the total Chern class of E by

c(E) := c_{0}(E) + c_{1}(E) + ⋯ + c_{r}(E)

viewed as an endomorphism of A_{∗}(X, K_{∗}). If E is the zero bundle (of rank 0), we set c_{0}(E) = 1 and c_{i}(E) = 0 if i ≠ 0.

Proposition 54.3. If E is a line bundle then c_{1}(E) = e(E).

Proof. We have ℙ(E) = X and L = E by Example 104.19. Therefore the equality (54.1) reads e(E)(α) − α_{1} = 0, hence c_{1}(E)(α) = α_{1} = e(E)(α). □

Example 54.4. If L is a line bundle, then c(L) = 1 + e(L). In particular, c(𝟙) = 1 by Proposition 53.5.

Proposition 54.5. Let f : Y → X be a morphism and E a vector bundle over X. Set E′ = f^{∗}(E). Then

(1) If f is proper then c(E) ∘ f_{∗} = f_{∗} ∘ c(E′).

(2) If f is flat then f^{∗} ∘ c(E) = c(E′) ∘ f^{∗}.

Proof. Let rank E = r. Consider the fiber product diagram

ℙ(E′) ⟶^{h} ℙ(E)
q′↓  ↓^{q}
Y ⟶^{f} X

with flat morphisms q and q′ of constant relative dimension r − 1. Denote by e and e′ the Euler classes of the tautological line bundle L over ℙ(E) and L′ over ℙ(E′), respectively. Note that L′ = h^{∗}(L).

(1): By Proposition 49.20, we have h_{∗} ∘ (q′)^{∗} = q^{∗} ∘ f_{∗}. By the definition of Chern classes, for every α′ ∈ A_{∗}(Y, K_{∗}) and α′_{i} = c_{i}(E′)(α′), we have:

∑_{i=0}^{r}(−1)^{i}(e′)^{r−i}(q′^{∗}(α′_{i})) = 0.

Applying h_{∗}, by Propositions 49.20 and 53.3(1), we have

0 = h_{∗} (∑_{i=0}^{r}(−1)^{i}(e′)^{r−i}(q′^{∗}(α′_{i})))

= ∑_{i=0}^{r}(−1)^{i}e^{r−i}(h_{∗}q′^{∗}(α′_{i}))

= ∑_{i=0}^{r}(−1)^{i}e^{r−i}(q^{∗}f_{∗}(α′_{i})).

Hence c_{i}(E)(f_{∗}(α′)) = f_{∗}(α′_{i}) = f_{∗}c_{i}(E′)(α′).

(2): By the definition of the Chern classes, for every α ∈ A_{∗}(X, K_{∗}) and α_{i} = c_{i}(E)(α) we have

∑_{i=0}^{r}(−1)^{i}e^{r−i}(q^{∗}α_{i}) = 0.

Applying h^{∗}, by Proposition 53.3(2), we have

0 = h^{∗}( ∑_{i=0}^{r}(−1)^{i}e^{r−i}(q^{∗}(α_{i})))

= ∑_{i=0}^{r}(−1)^{i}(e′)^{r−i}(h^{∗}q^{∗}(α_{i}))

= ∑_{i=0}^{r}(−1)^{i}(e′)^{r−i}(q′^{∗}f^{∗}(α_{i})).

Hence c_{i}(E′)(f^{∗}(α)) = f^{∗}(α_{i}) = f^{∗}c_{i}(E)(α). □

Proposition 54.6. Let E be a vector bundle over X possessing a filtration by subbundles with factors line bundles L_{1}, L_{2}, …, L_{r}. Then for every i = 1, …, r, we have

c_{i}(E) = σ_{i}(e(L_{1}), …, e(L_{r}))

where σ_{i} is the ith elementary symmetric function, i.e.,

c(E) = ∏_{i=1}^{r}(1 + e(L_{i})) = ∏_{i=1}^{r} c(L_{i}).

Proof. Let q : ℙ(E) → X be the canonical morphism and let e be the Euler class of the tautological line bundle L over ℙ(E). It follows from formula (54.1) and Proposition 54.5 that it suffices to prove that

∏_{i=1}^{r}(e − e(q^{∗}L_{i})) = 0

as an operation on A_{∗}(ℙ(E), K_{∗}). We induct on r. The case r = 1 follows from the fact that the tautological bundle L coincides with E over ℙ(E) = X (cf. Example 104.19). In the general case, let E′ be a subbundle of E having a filtration by subbundles with factors line bundles L_{1}, L_{2}, …, L_{r−1} and with E/E′ ≃ L_{r}. Consider the natural morphism f : U = ℙ(E) \ ℙ(E′) → ℙ(L_{r}). Under the identification of ℙ(L_{r}) with X, the bundle L_{r} is the tautological line bundle over ℙ(L_{r}). Hence f^{∗}(L_{r}) is isomorphic to the restriction of L to U. In other words, L|_{U} ≃ q^{∗}(L_{r})|_{U} and therefore e(L|_{U}) = e(q^{∗}(L_{r})|_{U}). It follows from Proposition 53.3 that for every α ∈ A_{∗}(ℙ(E), K_{∗}), we have

(e − e(q^{∗}L_{r}))(α)|_{U} = (e(L|_{U}) − e(q^{∗}(L_{r})|_{U}))(α|_{U}) = 0.

By the exactness of the localization sequence (52.6), there is an element β ∈ A_{∗}(ℙ(E′), K_{∗}) satisfying

i_{∗}(β) = (e − e(q^{∗}L_{r}))(α),

where i : ℙ(E′) → ℙ(E) is the closed embedding. Let L′ be the the tautological line bundle over ℙ(E′) and q′ : ℙ(E′) → X the canonical morphism. We have q′ = q ∘ i.

By induction and Proposition 53.3,

∏_{i=1}^{r}(e − e(q^{∗}L_{i}))(α) = ∏_{i=1}^{r−1}(e − e(q^{∗}L_{i}))(i_{∗}β)

= i_{∗} (∏_{i=1}^{r−1}(e(L′) − e(i^{∗}q^{∗}L_{i}))(β))

= i_{∗} (∏_{i=1}^{r−1}(e(L′) − e(q′^{∗}L_{i}))(β))
= 0.  □

Proposition 54.7 (Whitney Sum Formula). Let 0 → E′ ⟶^{f} E ⟶^{g} E′′ → 0 be an exact sequence of vector bundles over X. Then c(E) = c(E′) ∘ c(E′′), i.e.,

c_{n}(E) = ∑_{i+j=n} c_{i}(E′) ∘ c_{j}(E′′)

for every n.

Proof. By the Splitting Principle 53.13 and Proposition 54.5(2), we may assume that E′ and E′′ have filtrations by subbundles with quotients line bundles L′_{1}, …, L′_{r} and L′′_{1}, …, L′′_{s}, respectively. Consequently, E has a filtration with factors L′_{1}, …, L′_{r}, L′′_{1}, …, L′′_{s}. It follows from Proposition 54.6 that

c(E′) ∘ c(E′′) = ∏_{i=1}^{r} c(L′_{i}) ∘ ∏_{j=1}^{s} c(L′′_{i}) = c(E).  □

The same proof as for Corollary 53.2 yields:

Corollary 54.8. The Chern classes of any two vector bundles E and E′ over X commute: c(E′) ∘ c(E) = c(E) ∘ c(E′).

By Example 54.4, we have

Corollary 54.9. If E is a vector bundle over X, then c(E ⊕ 𝟙) = c(E). In particular, if E is a trivial vector bundle, then c(E) = 1.

The Whitney Sum Formula 54.7 allows us to define Chern classes not only for vector bundles over a scheme X but also for elements of the Grothendieck group K_{0}(X) of the category of vector bundles over X. Note that for a vector bundle E over X the endomorphisms c_{i}(E) are nilpotent for i > 0; therefore, the total Chern class c(E) is an invertible endomorphism. By the Whitney Sum Formula, the assignment E ↦ c(E) ∈ Aut(A_{∗}(X, K_{∗})) gives rise to the total Chern class homomorphism

c : K_{0}(X) → Aut(A_{∗}(X, K_{∗})).

55. Gysin and pull-back homomorphisms

In this section we consider contravariant properties of K-homology.

55.A. Gysin homomorphisms. Let f : Y → X be a regular closed embedding of codimension r and let p_{f} : N_{f} → Y be the canonical morphism (cf. §104.B). We define the Gysin homomorphism as the composition

f^{⋆} : A_{∗}(X, K_{∗}) ⟶^{σ_{f}} A_{∗}(N_{f}, K_{∗}) ⟶^{(p_{f}^{∗})^{−1}} A_{∗−r}(Y, K_{∗+r}).

Proposition 55.1. Let Z ⟶^{g} Y ⟶^{f} X be regular closed embeddings. Then (f ∘ g)^{⋆} = g^{⋆} ∘ f^{⋆}.

Proof. The normal bundles of the regular closed embeddings i : N_{f}|_{Z} → N_{f} and j : N_{g} → N_{f∘g} are canonically isomorphic, denote them by N. Consider the diagram

C_{∗}(X) ⟶^{σ_{f∘g}} C_{∗}(N_{f∘g}) ⟵^{p_{fg}^{∗}} C_{∗}(Z)
σ_{f} ↓  ^{σ_{j}} ↓  ∥
C_{∗}(N_{f}) ⟶^{σ_{i}} C_{∗}(N) ⟵^{(p_{g}p_{j})^{∗}} C_{∗}(Z)
p_{f}^{∗} ↑  ^{p_{j}^{∗}}↑  ∥
C_{∗}(Y) ⟶^{σ_{g}} C_{∗}(N_{g}) ⟵^{p_{g}^{∗}} C_{∗}(Z).

The bottom right square is commutative by Proposition 49.18. The bottom left and upper right squares are commutative by Proposition 51.5 and Lemma 51.9, respectively. The upper left square is commutative up to homotopy by Lemma 51.10. The statement follows from commutativity of the diagram up to homotopy.

□

Let

(55.2)  \begin{matrix}Y′ & ⟶^{f′} & X′ \\ ^{g}↓ & & ↓^{h} \\ Y & ⟶^{f} & X \end{matrix}

be a fiber product diagram with f and f′ regular closed embeddings. The natural morphisms i : N_{f′} → g^{∗}(N_{f}) of normal bundles over Y′ are closed embeddings. The factor bundle E = g^{∗}(N_{f})/N_{f′} over Y′ is called the excess vector bundle.

Proposition 55.3 (Excess Formula). Let h be a proper morphism. Then in the notation of diagram (55.2),

f^{⋆} ∘ h_{∗} = g_{∗} ∘ e(E) ∘ f′^{⋆}.

Proof. Let

p : N_{f} → Y, p′ : N_{f′} → Y′, i : N_{f′} → g^{∗}(N_{f}),
r : g^{∗}(N_{f}) → N_{f} and t : g^{∗}(N_{f}) → Y′

be the canonical morphisms. It suffices to prove that the diagram

\xymatrix{C_{∗}(X′) \ar[dd]_{h_{∗}} \ar[r]^{σ_{f′}} & C_{∗}(N_{f′}) \ar[dr]^{i_{∗}} \ar[dd]_{(ri)_{∗}} & C_{∗}(Y′) \ar[l]_{^{p′^{∗}}} \ar[dr]^{e(E)} \\  &  & C_{∗}(g^{∗}(N_{f})) \ar@{-->}[dl]_{r_{∗}} & C_{∗}(Y′) \ar[l]_{t^{∗}} \ar[dl]_{g_{∗}} \\ C_{∗}(X) \ar[r]^{σ_{f}} & C_{∗}(N_{f}) & C_{∗}(Y) \ar[l]_{p^{∗}}}

commutes.

The commutativity everywhere but at the top parallelogram follows by Propositions 49.20 and 51.6. Hence it suffices to show that t^{∗} ∘ e(E) = i_{∗} ∘ p′^{∗}. Consider the fiber product diagram

N_{f′} ⟶^{i} g^{∗}(N_{f})
p′↓  ↓^{j}
X ⟶^{s} E,

where j is the natural morphism of vector bundles and s is the zero section. Let q : E → Y′ be the natural morphism. It follows from the equality q ∘ j = t and Proposition 49.20 that

t^{∗} ∘ e(E) = t^{∗} ∘ q^{∗}^{−1} ∘ s_{∗} = j^{∗} ∘ s_{∗} = i_{∗} ∘ p′^{∗}.  □

Corollary 55.4. Suppose under the conditions of Proposition 55.3 that f and f′ are regular closed embeddings of the same codimension. Then f^{⋆} ∘ h_{∗} = g_{∗} ∘ f′^{⋆}.

Proof. In this case, E = 0 so e(E) is the identity. □

A consequence of Propositions 49.18 and 51.5 is the following:

Proposition 55.5. Suppose in the diagram (55.2) that h is a flat morphism. Then the diagram

A_{∗}(X, K_{∗}) ⟶^{f^{⋆}} A_{∗}(Y, K_{∗})
h^{∗}↓  ↓^{g^{∗}}
A_{∗}(X′, K_{∗}) ⟶^{f′^{⋆}} A_{∗}(Y′, K_{∗})

is commutative.

Proposition 55.6. Let f : Y → X be a regular closed embedding of equidimensional schemes. Then f^{⋆}([X]) = [Y].

Proof. By Example 49.13 and Proposition 52.7,

f^{⋆}([X]) = (p_{f}^{∗})^{−1} ∘ σ_{f}([X]) = (p_{f}^{∗})^{−1}([N_{f}]) = [Y].  □

Lemma 55.7. Let i : U → V and g : V → W be a regular closed embedding and a flat morphism, respectively, and let h = g ∘ i. If h is flat, then h^{∗} = i^{⋆} ∘ g^{∗}.

Proof. Let p : N_{i} → U be the canonical morphism. By Lemma 51.9, we have σ_{i} ∘ g^{∗} = (h ∘ p)^{∗} = p^{∗} ∘ h^{∗}, hence i^{⋆} ∘ g^{∗} = (p^{∗})^{−1} ∘ σ_{i} ∘ g^{∗} = h^{∗}. □

We turn to the study of the functorial behavior of Euler and Chern classes under Gysin homomorphisms. The next proposition is a consequence of Corollary 55.4 and Proposition 55.5 (cf. the proof of Proposition 53.3).

Proposition 55.8. Let f : Y → X be a regular closed embedding and L a line bundle over X. Set L′ = f^{∗}(L). Then f^{⋆} ∘ e(L) = e(L′) ∘ f^{⋆}.

As is in the proof of Proposition 54.5, we get

Proposition 55.9. Let f : Y → X be a regular closed embedding and E a vector bundle over X. Set E′ = f^{∗}(E). Then f^{⋆} ∘ c(E) = c(E′) ∘ f^{⋆}.

Proposition 55.10. Let f : Y → X be a regular closed embedding. Then f^{⋆}∘f_{∗} = e(N_{f}).

Proof. Let p : N_{f} → Y and s : Y → N_{f} be the canonical morphism and the zero section of the normal bundle respectively. By Corollary 51.7,

f^{⋆} ∘ f_{∗} = (p^{∗})^{−1} ∘ σ_{f} ∘ f_{∗} = (p^{∗})^{−1} ∘ s_{∗} = e(N_{f}).  □

Proposition 55.11. Let f : Y → X be a closed embedding given by a sheaf of locally principal ideals I ⊂ O_{X}. Let f′ : Y′ → X be the closed embedding given by the sheaf of ideals I^{n} for some n > 0 and g : Y → Y′ the canonical morphism. Then

f′^{⋆} = n(g_{∗} ∘ f^{⋆}).

Proof. We define a natural finite morphism h : D_{f} → D_{f′} of deformation schemes as follows: We may assume that X is affine, X = Spec(A), and Y = Spec(A/I). Then D_{f} = Spec(Ã) and D_{f′} = Spec(Ã′) where

Ã = ∐_{k∈ℤ} I^{−k}t^{k},  Ã′ = ∐_{k∈ℤ} I^{−kn}(t′)^{k}

(cf. §104.E).

The morphism h is induced by the ring homomorphism Ã′ → Ã taking a component I^{−kn}t′^{k} identically to I^{−kn}t^{kn}. In particular, the image of t′ is equal to t^{n}.

The morphism h yields a commutative diagram (cf. §104.E)

N_{f} ⟶ D_{f} ⟵ 𝔾_{m} × X
r↓  ^{h}↓  ^{q}↓
N_{f′} ⟶ D_{f′} ⟵ 𝔾_{m} × X,

where q is the identity on X and the nth power morphism on 𝔾_{m}. Let ∂ (respectively, ∂′) be the boundary map with respect to the top row (respectively, the bottom row) of the diagram. It follows from Proposition 49.36(1) that

(55.12)  r_{∗} ∘ ∂ = ∂′ ∘ q_{∗}.
For any α ∈ C_{∗}(X) we have
(55.13)  q_{∗}({t} · ([𝔾_{m}] × α)) = {±t′} · ([𝔾_{m}] × α)

since the norm of t in the field extension F (t)/F (t′) is equal to ±t′. By (55.12) and (55.13), we have

(r_{∗} ∘ σ_{f})(α) = (r_{∗} ∘ ∂)({t} · ([𝔾_{m}] × α))
= (∂′ ∘ q_{∗})({t} · ([𝔾_{m}] × α))
= ∂′({±t′} · ([𝔾_{m}] × α))
= σ_{f′}(α),

hence

(55.14)  r_{∗} ∘ σ_{f} = σ_{f′}.

The morphism p factors into the composition of morphisms in the first row of the commutative diagram

N_{f} ⟶^{i} (N_{f})^{⊗n} ⟶^{j} N_{f′}
p↓  ^{s}↓  ^{p′}↓
Y  Y ⟶^{g} Y′

of vector bundles. As the morphism i is finite flat of degree n, the composition i_{∗} ∘ i^{∗} is multiplication by n. The right square of the diagram is a fiber square. Hence by Proposition 49.20, we have

r_{∗} ∘ p^{∗} = j_{∗} ∘ i_{∗} ∘ i^{∗} ∘ s^{∗} = n(j_{∗} ∘ s^{∗}) = n(p′^{∗} ∘ g_{∗}).

It follows from (55.14) that

f′^{⋆} = (p′^{∗})^{−1} ∘ σ_{f′} = (p′^{∗})^{−1} ∘ r_{∗} ∘ σ_{f} = n(g_{∗} ∘ (p^{∗})^{−1} ∘ σ_{f}) = n(g_{∗} ∘ f^{⋆})

as needed. □

55.B. The pull-back homomorphisms. Let f : Y → X be a morphism of equidimensional schemes with X smooth. By Corollary 104.14, the morphism

i = (1_{Y} , f) : Y → Y × X

is a regular closed embedding of codimension d_{X} = dim X with the normal bundle N_{i} = f^{∗}(T_{X}), where T_{X} is the tangent bundle of X (cf. Corollary 104.14). The projection p : Y × X → X is a flat morphism of relative dimension d_{Y}. Set d = d_{X} − d_{Y}. We define the pull-back homomorphism

(55.15)  f^{∗} : A_{∗}(X, K_{∗}) → A_{∗−d}(Y, K_{∗+d})

as the composition i^{⋆} ∘ p^{∗}.

We use the same notation for the pull-back homomorphism just defined and the flat pull-back. The following proposition justifies this notation.

Proposition 55.16. Let f : Y → X be a flat morphism of equidimensional schemes and let X be smooth. Then the pull-back f^{∗} in (55.15) coincides with the flat pullback homomorphism.

Proof. This follows by applying Lemma 55.7 to the closed embedding i = (1_{Y} , f) : Y → Y × X and to the projection g : Y × X → X. □

We have the following two propositions about the compositions of the pull-back maps.

Proposition 55.17. Let Z ⟶^{g} Y ⟶^{f} X be morphisms of equidimensional schemes with X smooth and g flat. Then (f ∘ g)^{∗} = g^{∗} ∘ f^{∗}.

Proof. Consider the fiber product diagram

Z ⟶^{g} Y
i_{Z} ↓  ↓^{i_{Y}}
Z × X ⟶^{h} Y × X,

where i_{Y} = (1_{Y} , f), i_{Z} = (1_{Z}, fg), h = (g, 1_{X}) and two projections p_{Y} : Y × X → X and p_{Z} : Z × X → X. We have p_{Z} = p_{Y} ∘ h. By Propositions 49.18 and 55.5,

(f ∘ g)^{∗} = i_{Z}^{⋆} ∘ p_{Z}^{∗}
= i_{Z}^{⋆} ∘ h^{∗} ∘ p_{Y}^{∗}
= g^{∗} ∘ i_{Y}^{⋆} ∘ p_{Y}^{∗}
= g^{∗} ∘ f^{∗}.  □

Proposition 55.18. Let Z ⟶^{g} Y ⟶^{f} X be morphisms of equidimensional schemes with both Y and X smooth. Then (f ∘ g)^{∗} = g^{∗} ∘ f^{∗}.

Proof. Consider the commutative diagram

\xymatrix{Z \ar[dr]_{k} \ar[r]^{(1_{Z},g)} \ar[dd]_{(1_{Z},fg)} & Z × Y \ar[r] \ar[d]^{h} & Y \ar[d]^{(1_{Y} ,f)} \\  & Z × Y × X \ar[dr] \ar[r] & Y × X \ar[d] \\ Z × X \ar[ur]^{l} \ar[rr] &  & X}

where

k = (1_{Z}, g, fg), h(z, y) = (z, y, f(y)), l(z, x) = (z, g(z), x),

and all unmarked arrows are the projections. Applying the Gysin homomorphisms or the flat pull-backs for all arrows in the diagram, we get a diagram of homomorphisms of the K-homology groups that is commutative by Propositions 49.18, 55.1, 55.5, and Lemma 55.7. □

We next show that the pull-back homomorphism for a regular closed embedding coincides with the Gysin homomorphism:

Proposition 55.19. Let f : Y → X be a regular closed embedding of equidimensional schemes with X smooth. Then f^{∗} = f^{⋆}.

Proof. The commutative diagram

\xymatrix{Y \ar[r]^{d} \ar[dr]_{1_{Y}} & Y × Y \ar[r]^{h} \ar[d]^{p} & Y × X \ar[d]^{q} \\  & Y \ar[r]^{f} & X,}

with d the diagonal embedding and h = 1_{Y} × f gives rise to the diagram

\xymatrix{A_{∗}(X, K_{∗}) \ar[r]^{f^{⋆}} \ar[d]_{q^{∗}} & A_{∗}(Y, K_{∗}) \ar[d]_{p^{∗}} \ar[dr]^{1} \\ A_{∗}(Y × X, K_{∗}) \ar[r]^{h^{⋆}} & A_{∗}(Y × Y, K_{∗}) \ar[r]^{d^{⋆}} & A_{∗}(Y, K_{∗}).}

The square is commutative by Proposition 55.5 and the triangle by Lemma 55.7. Let g = h ∘ d. Then

f^{∗} = g^{⋆} ∘ q^{∗} = d^{⋆} ∘ h^{⋆} ∘ q^{∗} = f^{⋆}.  □

The pull-back homomorphisms commute with external products:

Proposition 55.20. Let f : X′ → X and g : Y′ → Y be morphisms of equidimensional schemes with X and Y smooth. Then for every α ∈ C_{∗}(X) and β ∈ C_{∗}(Y), we have

(f × g)^{∗}(α × β) = f^{∗}(α) × g^{∗}(β).

Proof. It suffices to do the case with g = 1_{Y} and by Proposition 50.5 with f a regular closed embedding. Denote by q_{X} : 𝔾_{m} × X → X and p_{f} : N_{f} → X′ the canonical morphisms. Note that N_{f×1_{Y}} = N_{f} × Y. Consider the diagram

C_{∗}(X) ⟶^{q_{X}^{∗}} C_{∗}(𝔾_{m} × X) ⟶^{{t}} C_{∗}(𝔾_{m} × X)
↓  ↓  ↓
C_{∗}(X × Y) ⟶^{q_{X×Y}^{∗}} C_{∗}(𝔾_{m} × X × Y) ⟶^{{t}} C_{∗}(𝔾_{m} × X × Y)
C_{∗}(𝔾_{m} × X) ⟶^{∂} C_{∗}(N_{f}) ⟵^{p_{f}^{∗}}  C_{∗}(X′)
↓  ↓  ↓
C_{∗}(𝔾_{m} × X × Y) ⟶^{∂} C_{∗}(N_{f×1_{Y}}) ⟵^{(p_{f} ×1_{Y} )^{∗}} C_{∗}(X′ × Y)

where all vertical homomorphisms are given by the external product with β. The commutativity of all squares follow from Propositions 50.5, 50.7, and 50.8. □

Proposition 55.21. Let f : Y → X be a morphism of equidimensional schemes with X smooth. Then f^{∗}([X]) = [Y].

Proof. Let i = (1_{Y} , f) : Y → Y ×X be the graph of f and let p : Y ×X → X be the projection. It follows from Corollary 50.6 and Proposition 55.6 that

f^{∗}([X]) = i^{⋆} ∘ p^{∗}([X]) = i^{⋆}([Y × X]) = [Y].  □

The following statement on the commutativity of the pull-back maps and the Chern classes is a consequence of Propositions 54.5(2) and 55.9.

Proposition 55.22. Let f : Y → X be a morphism of equidimensional schemes with X smooth and E a vector bundle over X. Set E′ = f^{∗}(E). Then f^{∗} ∘ c(E) = c(E′) ∘ f^{∗}.

56. K-COHOMOLOGY RING OF SMOOTH SCHEMES 257

56. K-cohomology ring of smooth schemes

We now consider the case that our scheme X is smooth. This allows us to introduce the K-cohomology groups A^{∗}(X, K_{∗}) which we do as follows: If X is irreducible of dimension d, we set

A^{p}(X, K_{q}) := A_{d−p}(X, K_{q−d}).

In the general case, let X_{1}, X_{2}, …, X_{s} be (disjoint) irreducible components of X. We set

A^{p}(X, K_{q}) := ∐_{i=1}^{s} A^{p}(X_{i}, K_{q}).

In particular, if X is an equidimensional smooth scheme of dimension d, then A^{p}(X, K_{q}) = A_{d−p}(X, K_{q−d}).

Let f : Y → X be a morphism of smooth schemes. We define the pull-back homomorphism

f^{∗} : A^{p}(X, K_{q}) → A^{p}(Y, K_{q})

as follows: If X and Y are both irreducible and of dimension d_{X} and d_{Y}, respectively, we define f^{∗} as in §55.B:

f^{∗} : A^{p}(X, K_{q}) = A_{d_{X}−p}(X, K_{q−d_{X}}) ⟶^{f^{∗}} A_{d_{Y} −p}(X, K_{q−d_{Y}}) = A^{p}(Y, K_{q}).

If just Y is irreducible, we have f(Y) ⊂ X_{i} for an irreducible component X_{i} of X. We define the pull-back as the composition

A^{p}(X, K_{q}) → A^{p}(X_{i}, K_{q}) ⟶^{f^{∗}} A^{p}(Y, K_{q}),

where the first map is the canonical projection. Finally, in the general case, we define f^{∗} as the direct sum of the homomorphisms A^{p}(X, K_{q}) → A^{p}(Y_{j}, K_{q}) over all irreducible components Y_{j} of Y.

It follows from Proposition 55.18 that if Z ⟶^{g} Y ⟶^{f} X are morphisms of smooth schemes, then (f ∘ g)^{∗} = g^{∗} ∘ f^{∗}.

Let X be a smooth scheme. Denote by

d = d_{X} : X → X × X

the diagonal closed embedding. The composition

(56.1) A^{p}(X, K_{q}) ⊗ A^{p′}(X, K_{q′}) ⟶^{×} A^{p+p′}(X × X, K_{q+q′}) ⟶^{d^{∗}} A^{p+p′}(X, K_{q+q′})

defines a product on A^{∗}(X, K_{∗}).

Remark 56.2. If X = X_{1} ∐ X_{2}, then A^{∗}(X, K_{∗}) = A^{∗}(X_{1}, K_{∗}) ⊕ A^{∗}(X_{2}, K_{∗}). Since the image of the diagonal morphism d_{X} does not intersect X_{1} × X_{2}, the product of two classes from A^{∗}(X_{1}, K_{∗}) and A^{∗}(X_{2}, K_{∗}) is zero.

Proposition 56.3. The product in (56.1) is associative.

Proof. Let α, β, γ ∈ A^{∗}(X, K_{∗}). By Proposition 55.20, we have
(α × β) × γ =d^{∗}(d^{∗}(α × β) × γ)
= d^{∗} ∘ (d × 1_{X})^{∗}(α × β × γ)
= ((d × 1_{X}) ∘ d)^{∗}(α × β × γ)
= c^{∗}(α × β × γ),

where c : X → X × X × X is the diagonal embedding. Similarly, α × (β × γ) = c^{∗}(α × β × γ). □

Proposition 56.4. For every smooth scheme X, the product in A^{∗}(X, K_{∗}) is bigraded commutative, i.e., if α ∈ A^{p}(X, K_{q}) and α′ ∈ A^{p′}(X, K_{q′}), then

α · α′ = (−1)^{(p+q)(p′+q′)}α′ · α.

Proof. It follows from (50.1) that

α · α′ = d^{∗}(α × α′) = (−1)^{(p+q)(p′+q′)}d^{∗}(α′ × α) = (−1)^{(p+q)(p′+q′)}α′ · α. □

Let X be a smooth scheme and let X_{1}, X_{2}, … be the irreducible components of X. Then [X] = ∑[X_{i}] in A^{0}(X, K_{0}).

Proposition 56.5. The class [X] is the identity in A^{∗}(X, K_{∗}) under the product.

Proof. We may assume that X is irreducible. Let f : X × X → X be the first projection. Since f ∘ d = 1_{X}, it follows from Corollary 50.6 and Proposition 55.16 that

α · [X] = d^{∗}(α × [X]) = d^{∗}f^{∗}(α) = α.  □

We have proven:

Theorem 56.6. Let X be a smooth scheme. Then A^{∗}(X, K_{∗}) is a bigraded commutative associative ring with the identity [X].

Remark 56.7. If X_{1}, …, X_{n} are the irreducible components of a smooth scheme X, the ring A^{∗}(X, K_{∗}) is the product of the rings A^{∗}(X_{1}, K_{∗}), …, A^{∗}(X_{n}, K_{∗}).

Proposition 56.8. Let f : Y → X be a morphism of smooth schemes. Then

f^{∗}(α · β) = f^{∗}(α) · f^{∗}(β)

for all α, β ∈ A^{∗}(X, K_{∗}) and f^{∗}([X]) = [Y].

Proof. Since (f × f) ∘ d_{Y} = d_{X} ∘ f, it follows from Propositions 55.18 and 55.20 that

f^{∗}(α · β) = f^{∗} ∘ d_{X}^{∗}(α × β)
= d_{Y}^{∗} ∘ (f × f)^{∗}(α × β)
= d_{Y}^{∗} (f(α) × f(β))
= f^{∗}(α) · f^{∗}(β).

The second equality follows from Proposition 55.21. □

It follows from Proposition 56.8 that the correspondence X ↦ A^{∗}(X, K_{∗}) gives rise to a cofunctor from the category of smooth schemes and arbitrary morphisms to the category of bigraded rings and homomorphisms of bigraded rings.

Proposition 56.9 (Projection Formula). Let f : Y → X be a proper morphism of smooth schemes. Then

f_{∗}(α · f^{∗}(β)) = f_{∗}(α) · β

for every α ∈ A^{∗}(Y, K_{∗}) and β ∈ A^{∗}(X, K_{∗}).

Proof. Let g = (1_{Y} × f) ∘ d_{Y}. Then we have the fiber product diagram

(56.10)  \begin{matrix}Y & ⟶^{g} & Y × X \\ ^{f}↓ & & ↓^{f×1_{X}} \\ X & ⟶^{d_{X}} & X × X. \end{matrix}

It follows from Propositions 49.20 and 55.20 that

f_{∗}(α · f^{∗}(β)) =f_{∗} ∘ d_{Y}^{∗} (α × f^{∗}(β))
= f_{∗} ∘ d_{Y}^{∗} ∘ (1_{Y} × f)^{∗}(α × β)
= f_{∗} ∘ g^{∗}(α × β)
= d_{X}^{∗} ∘ (f × 1_{Y})_{∗}(α × β)
= d_{X}^{∗}(f_{∗}(α) × β)
= f_{∗}(α) · β.  □

The projection formula asserts that the push-forward homomorphism f_{∗} is A^{∗}(X, K_{∗})-linear if we view A^{∗}(Y, K_{∗}) as a A^{∗}(X, K_{∗})-module via f^{∗}.

The following statement is an analog of the projection formula.

Proposition 56.11. Let f : Y → X be a morphism of equidimensional schemes with X smooth. Then

f_{∗}(f^{∗}(β)) = f_{∗}([Y]) · β

for every β ∈ A^{∗}(X, K_{∗}).

Proof. The closed embeddings g and d_{X} in the diagram (56.10) are regular of the same codimension (cf. Corollary 104.14). Let p : X × X → X be the second projection. Then the composition q = p ∘ (f × 1_{X}) : Y × X → X is also the projection. By Propositions 50.4, 50.5, 55.17, 55.21, and Corollaries 50.6, 55.4, we have

f_{∗}(f^{∗}(β)) =f_{∗} ∘ g^{⋆} ∘ q^{∗}(β)
= f_{∗} ∘ g^{⋆} ∘ (f × 1_{X})^{∗} ∘ p^{∗}(β)
= d_{X}^{⋆} ∘ (f × 1_{X})_{∗} ∘ (f × 1_{X})^{∗} ∘ p^{∗}(β)
= d_{X}^{⋆} ∘ (f_{∗} × id) ∘ (f^{∗} × id)([X] × β)
= d_{X}^{⋆}(f_{∗} ∘ f^{∗}([X]) × β)
= d_{X}^{⋆}(f_{∗}([Y]) × β)
= f_{∗}([Y]) · β.  □

CHAPTER X Chow Groups

In this chapter we study Chow groups as special cases of K-homology and K-cohomology theories, so we can apply results from the previous chapter. Chow groups will remain the main tool in the rest of the book. We also develop the theory of Segre classes that will be used in the chapter on the Steenrod operations that follows.

57. Definition of Chow groups

Recall that a scheme is a separated scheme of finite type over a field and a variety is an integral scheme.

57.A. Two equivalent definitions of Chow groups.

Definition 57.1. Let X be a scheme over F and p ∈ ℤ. We call the group

CH_{p}(X) := A_{p}(X, K_{−p})

the Chow group of dimension p classes of cycles on X.

By definition,

CH_{p}(X) = Coker ( ∐_{x∈X_{(p+1)}} K_{1}(F (x)) ⟶^{d_{X}} ∐_{x∈X_{(p)}} K_{0}(F (x))) .

Note that K_{1}(F (x)) = F (x)^{×} and K_{0}(F (x)) = ℤ. Thus the Chow group CH_{p}(X) is the factor group of the free abelian group

Z_{p}(X) = ∐_{x∈X_{(p)}} ℤ,

called the group of p-dimensional cycles on X, by the subgroup generated by the divisors d_{X}(f) = div(f) for all f ∈ F (x)^{×} and x ∈ X_{(p+1)}.

A point x ∈ X of dimension p gives rise to a prime cycle in Z_{p}(X), denoted by [x]. Thus, an element of Z_{p}(X) is a finite formal linear combination ∑ n_{x}[x] with n_{x} ∈ ℤ and dim x = p. We will often write \overline{\{x\}} instead of x, so that an element of Z_{p}(X) is a finite formal linear combination ∑ n_{Z}[Z] where the sum is taken over closed subvarieties Z ⊂ X of dimension p. We will use the same notation for the classes of cycles in CH_{p}(X). Note that a closed subscheme W ⊂ X (not necessarily integral) defines the cycle [W] ∈ Z(X) (cf. Example 49.2).

Example 57.2. Let X be a scheme of dimension d. The group CH_{d}(X) = Z_{d}(X) is free with basis the classes of irreducible components (generic points) of X of dimension d.

The divisor of a function can be computed in a simpler way. Let R be a 1dimensional Noetherian local domain with quotient field L. We define the order homomorphism

ord_{R} : L^{×} → ℤ

by the formula ord_{R}(r) = l(R/rR) for every nonzero r ∈ R.

Let Z be a variety over F of dimension d. For any point x ∈ Z of dimension d − 1, the local ring O_{Z,x} is 1-dimensional. Hence the order homomorphism

ord_{x} = ord_{O_{Z,x}} : F (Z)^{×} → ℤ

is well-defined.

Proposition 57.3. Let Z be a variety over F of dimension d and f ∈ F (Z)^{×}. Then div(f) = ∑ ord_{x}(f) · x, where the sum is taken over all points x ∈ Z of dimension d − 1.

Proof. Let R be the local ring O_{Z,x}, where x is a point of dimension d − 1. Let R̃ denote the integral closure of R in F (Z). For every nonzero f ∈ R, the x-component of div(f) is equal to

∑ l(R̃_{Q}/fR̃_{Q}) · [R̃/Q : F (x)],

where the sum is taken over all maximal ideals Q of R̃. Applying Lemma 102.3 to the R̃-module M = R̃/fR̃, we have the x-component equals l_{R}(R̃/fR̃). Since R̃/R is an R-module of finite length, l_{R}(R̃/fR̃) = l_{R}(R/fR) = ord_{x}(f). □

We wish to give an equivalent definition of Chow groups. To do so we need some preliminaries.

Let Z be a variety over F of dimension d and f : Z → ℙ^{1} a dominant morphism. Thus f is a flat morphism of relative dimension d−1. For any rational point a ∈ ℙ^{1}, the pull-back scheme f^{−1}(a) is an equidimensional subscheme of Z of dimension d − 1. Note that we can view f as a rational function on Z.

Lemma 57.4. Let f be as above. Then div(f) = [f^{−1}(0)] − [f^{−1}(∞)] on Z.

Proof. Let x ∈ Z be a point of dimension d − 1 with the x-component of div(f) nontrivial. Then f(x) = 0 or f(x) = ∞.

Consider the first case, so f ∈ O_{Z,x}. By Proposition 57.3, the x-component of div(f) is equal to ord_{x}(f). The local ring O_{f^{−1}(0),x} coincides with O_{Z,x}/fO_{Z,x}, therefore, the x-component of [f^{−1}(0)] is equal to

l(O_{f^{−1}(0),x}) = l(O_{Z,x}/fO_{Z,x}) = ord_{x}(f).

Similarly (applying the above argument to the function f^{−1}), we see that in the second case the x-component of [f^{−1}(∞)] is equal to ord_{x}(f^{−1}) = − ord_{x}(f). □

Let X be a scheme and Z ⊂ X × ℙ^{1} a closed subvariety of dimension d with Z dominant over ℙ^{1}. Hence the projection f : Z → ℙ^{1} is flat of relative dimension d − 1. For every rational point a ∈ ℙ^{1}, the projection p : X × ℙ^{1} → X maps the subscheme f^{−1}(a) isomorphically onto a closed subscheme of X that we denote by Z(a). It follows from Lemma 57.4 that

(57.5)  p_{∗}(div(f)) = [Z(0)] − [Z(∞)].

In particular, the classes of [Z(0)] and [Z(∞)] coincide in CH(X).

Let Z(X; ℙ^{1}) denote the subgroup of Z(X×ℙ^{1}) generated by the classes of closed subvarieties of X × ℙ^{1} that are dominant over ℙ^{1}. For any cycle β ∈ Z(X; ℙ^{1}) and any rational point a ∈ ℙ^{1}, the cycle β(a) ∈ Z(X) is well-defined.

If α = ∑ n_{Z}[Z] ∈ Z(X), we write α×[ℙ^{1}] for the cycle ∑ n_{Z}[Z×ℙ^{1}] ∈ Z(X; ℙ^{1}). Clearly, (α × [ℙ^{1}])(a) = α for any rational point a ∈ ℙ^{1}.

Let α and α′ be two cycles on a scheme X. We say that α and α′ are rationally equivalent if the classes of α and α′ are equal in CH(X).

Proposition 57.6. Two cycles α and α′ on a scheme X are rationally equivalent if and only if there is a cycle β ∈ Z(X; ℙ^{1}) satisfying α = β(0) and α′ = β(∞).

Proof. It was shown in (57.5) that the classes of the cycles β(0) and β(∞) are equal in CH(X). Conversely, suppose that the classes of α and α′ are equal in CH(X). By the definition of the Chow group, there are closed subvarieties Z_{i} ⊂ X and nonconstant rational functions g_{i} on Z_{i} such that

α − α′ = ∑ div(g_{i}).

Let V_{i} be closure of the graph of g_{i} in Z_{i} × ℙ^{1} ⊂ X × ℙ^{1} and let f_{i} : V_{i} → ℙ^{1} be the induced morphism. Since g_{i} is nonconstant, the morphism f_{i} is dominant and [V_{i}] ∈ Z(X; ℙ^{1}).

The projection p : X × ℙ^{1} → X maps V_{i} birationally onto Z_{i}, hence by Proposition 49.9,

div(g_{i}) = div(p_{∗}(f_{i})) = p_{∗} div(f_{i}) = [V_{i}(0)] − [V_{i}(∞)].

Let β′ = ∑[V_{i}] ∈ Z(X; ℙ^{1}). We have

α − α′ = β′(0) − β′(∞).

Consider the cycle

γ = α − β′(0) = α′ − β′(∞)

and set β′′ = γ × [ℙ^{1}] and β = β′ + β′′. Then β(0) = β′(0) + β′′(0) = β′(0) + γ = α and similarly β(∞) = α′. □

It follows from the above that an equivalent definition of the Chow group CH(X) is given as the factor group of the the group of cycles Z(X) modulo the subgroup of cycles of the form β(0) − β(∞) for all β ∈ Z(X; ℙ^{1}).

57.B. Functorial properties of the Chow groups. We now specialize the functorial properties developed in the previous chapter to Chow groups.

A proper morphism f : X → Y gives rise to the push-forward homomorphism

f_{∗} : CH_{p}(X) → CH_{p}(Y).

Example 57.7. Let X be a complete scheme over F. The push-forward homomorphism deg : CH(X) → CH(Spec F) = ℤ induced by the structure morphism X → Spec(F) is called the degree homomorphism. For any x ∈ X, we have

deg([x]) = \{ \begin{matrix}deg(x) = [F (x) : F] & if x is a closed point, \\ 0 & otherwise. \end{matrix}

A flat morphism g : Y → X of relative dimension d defines the pull-back homomorphism

g^{∗} : CH_{p}(X) → CH_{p+d}(Y).

Proposition 57.8. Let g : Y → X be a flat morphism of schemes over F of relative dimension d and W ⊂ X a closed subscheme of pure dimension k. Then g^{∗}([W]) = [g^{−1}(W)] in Z_{d+k}(Y).

Proof. Consider the fiber product diagram of natural morphisms

g^{−1}(W) ⟶^{f} W ⟶^{p} Spec(F)
j↓  ↓^{i}
Y  ⟶^{g} X.

By Propositions 49.18 and 49.20,

g^{∗}([W]) = g^{∗} ∘ i_{∗} ∘ p^{∗}(1) = j_{∗} ∘ f^{∗} ∘ p^{∗}(1) = j_{∗} ∘ (p ∘ f)^{∗}(1) = [g^{−1}(W)]. □

The localization property (cf. §52.D) yields:

Proposition 57.9 (Localization Sequence). Let X be a scheme, Z ⊂ X a closed subscheme and U = X \ Z. Let i : Z → X and j : U → X be the closed embedding and the open immersion respectively. Then the sequence

CH_{p}(Z) ⟶^{i_{∗}} CH_{p}(X) ⟶^{j^{∗}} CH_{p}(U) → 0

is exact.

The following proposition shows that the restriction on the generic fiber of a morphism is surjective on Chow groups.

Proposition 57.10. Let X be a variety of dimension n and f : Y → X a dominant morphism. Let x denote the generic point of X and Y_{x} the generic fiber of f. Then the pull-back homomorphism CH_{p}(Y) → CH_{p−n}(Y_{x}) is surjective.

Proof. By the continuity property (cf. Proposition 52.9), the pull-back homomorphism CH_{p}(Y) → CH_{p−n}(Y_{x}) is the colimit of surjective restriction homomorphisms CH_{p}(Y) → CH_{p}(f^{−1}(U)) over all nonempty open subschemes U of X and therefore is surjective. □

Corollary 57.11. For every variety X of dimension n and scheme Y over F, the pull-back homomorphism CH_{p}(X × Y) → CH_{p−n}(Y_{F (X)}) is surjective.

Let X and Y be two schemes. It follows from §52.C that there is a product map of Chow groups

CH_{p}(X) ⊗ CH_{q}(Y) → CH_{p+q}(X × Y).

Proposition 57.12. Let Z ⊂ X and W ⊂ Y be two closed equidimensional subschemes of dimensions d and e, respectively. Then

[Z × W] = [Z] × [W] in Z_{d+e}(X × Y).

Proof. Let p : Z → Spec(F) and q : W → Spec(F) be the structure morphisms and i : Z → X and j : W → Y the closed embeddings. By Example 49.2 and Propositions 50.4, 50.5, we have

[Z × W] = (i × j)_{∗} ∘ (p × q)^{∗}(1) = (i_{∗} ∘ p^{∗}(1)) × (j_{∗} ∘ q^{∗}(1)) = [Z] × [W]. □

Theorem 57.13 (Homotopy Invariance; cf. Theorem 52.13). Let g : Y → X be a flat morphism of schemes over F of relative dimension d. Suppose that for every x ∈ X, the fiber Y_{x} is isomorphic to the affine space 𝔸_{F (x)}^{d}. Then the pull-back homomorphism

g^{∗} : CH_{p}(X) → CH_{p+d}(Y)

is an isomorphism for every p.

Theorem 57.14 (Projective Bundle Theorem; cf. Theorem 53.10). Let E → X be a vector bundle of rank r, q : ℙ(E) → X the associated projective bundle morphism, and e the Euler class of the canonical or tautological line bundle over ℙ(E). Then the homomorphism

∐_{i=1}^{r} e^{r−i} ∘ q^{∗} : ∐_{i=1}^{r} CH_{∗−i+1}(X) → CH_{∗}(ℙ(E))

is an isomorphism, i.e., every α ∈ CH_{∗}(ℙ(E)) can be written in the form

α = ∑_{i=1}^{r} e^{r−i}(q^{∗}(α_{i}))

for uniquely determined elements α_{i} ∈ CH_{∗−i+1}(X).

Example 57.15. Let X = ℙ(V), where V is a vector space of dimension d+1 over F. For every p ∈ [0, d], let l_{p} ∈ CH_{p}(ℙ(V)) be the class of the subscheme ℙ(V_{p}) of X, where V_{p} is a subspace of V of dimension p + 1. By Corollary 53.7,

CH_{p}(ℙ(V)) = \{ \begin{matrix}ℤ · l_{p} & if 0 ≤ p ≤ d, \\ 0 & otherwise. \end{matrix}

Let f : Y → X be a regular closed embedding of codimension r. As usual we write N_{f} for the normal bundle of f. The Gysin homomorphism

f^{⋆} : CH_{∗}(X) → CH_{∗−r}(Y)

is defined by the formula f^{⋆} = (p^{∗})^{−1} ∘ σ_{f}, where p : N_{f} → Y is the canonical morphism and σ_{f} is the deformation homomorphism.

Let Z ⊂ X be a closed subscheme of pure dimension k and set W = f^{−1}(Z). The cone C_{g} of the restriction g : W → Z of f is of pure dimension k. Proposition 52.7 yields:

Corollary 57.16. Under the conditions of Proposition 52.7, we have f^{⋆}([Z]) = (p^{∗})^{−1}h_{∗}([C_{g}]).

Lemma 57.17. Let C′ be an irreducible component of C_{g}. Then C′ is an integral cone over a closed subvariety W′ ⊂ W with dim W′ ≥ k − r.

Proof. Let N′ be the restriction of the normal bundle N_{f} on W′. Since C′ is a closed subvariety of N′ of dimension k (cf. Example 104.3), we have

k = dim C′ ≤ dim N′ = dim W′ + r.  □

Corollary 57.18. Let V ⊂ W be an irreducible component. Then there is an irreducible component of C_{g} that is a cone over V. In particular, dim V ≥ k − r.

Proof. Let v ∈ V be the generic point. Since the canonical morphism q : C_{g} → W is surjective (that is, split by the zero section), there is an irreducible component C′ ⊂ C_{g} such that v ∈ Im q. Clearly, Im q = V, i.e., C′ is a cone over V. □

We say that the scheme Z has a proper inverse image with respect to f if every irreducible component of W = f^{−1}(Z) has dimension k − r.

Proposition 57.19. Let f : Y → X be a regular closed embedding of schemes over F of codimension r and Z ⊂ X a closed equidimensional subscheme having proper inverse image with respect to f. Let V_{1}, V_{2}, …, V_{s} be all the irreducible components of W = f^{−1}(Z), so [W] = ∑ n_{i}[V_{i}] for some n_{i} > 0. Then

f^{⋆}([Z]) = ∑_{i=1}^{s} m_{i}[V_{i}],

for some integers m_{i} with 1 ≤ m_{i} ≤ n_{i}.

Proof. Let g : W → Z be the restriction of f and C_{i} the restriction of the cone C_{g} on V_{i}. Let N_{i} be the restriction to V_{i} of the normal cone N_{f}. As N_{i} is a vector bundle of rank r over the variety V_{i} of dimension k − r, the variety N_{i} is of dimension k. Moreover, the N_{i} are all of the irreducible components of the restriction N of N_{f} to W and [N] = ∑ n_{i}[N_{i}].

The cone C_{g} is a closed subscheme of N of pure dimension k. Hence C_{i} is a closed subscheme of N_{i} of pure dimension k for each i. Since N_{i} is a variety of dimension k, the closed embedding of C_{i} into N_{i} is an isomorphism. In particular, the C_{i} are all of the irreducible components of C_{g}, so [C_{g}] = ∑ m_{i}[C_{i}] with m_{i} = l(O_{C_{g},x_{i}}) and where x_{i} ∈ C_{g} is the generic point of C_{i}. In view of Example 49.13, we have

h_{∗}([C_{i}]) = [N_{i}] = p^{∗}([V_{i}])

and by Corollary 57.16,

f^{⋆}([Z]) = (p^{∗})^{−1}h_{∗}([C_{g}]) = (p^{∗})^{−1} ∑ m_{i}h_{∗}([C_{i}]) = ∑ m_{i}[V_{i}].

Finally, the closed embedding h : C_{g} → N induces a surjective ring homomorphism O_{N,y_{i}} → O_{C_{g},x_{i}}, where y_{i} ∈ N is the generic point of N_{i}. Therefore,

1 ≤ m_{i} = l(O_{C_{g},x_{i}}) ≤ l(O_{N,y_{i}}) = n_{i}.  □

Corollary 57.20. Suppose the conditions of Proposition 57.19 hold and, in addition, the scheme W is reduced. Then f^{⋆}([Z]) = ∑[V_{i}], i.e., all the m_{i} = 1.

Proof. Indeed, all n_{i} = 1, hence all m_{i} = 1. □

Let Y and Z be closed subvarieties of a smooth scheme X of codimensions p and q, respectively. We say that Y and Z intersect properly if every component of Y ∩ Z has codimension p + q.

Applying Proposition 57.19 to the regular diagonal embedding X → X × X and the subscheme Y × Z yields the following:

Proposition 57.21. Let Y and Z be two closed subvarieties of a smooth scheme X that intersect properly. Let V_{1}, V_{2}, …, V_{s} be all irreducible components of W = Y ∩Z and [W] = ∑ n_{i}[V_{i}] for some n_{i} > 0. Then

[Y] · [Z] = ∑_{i=1}^{s} m_{i}[V_{i}],

for some integers m_{i} with 1 ≤ m_{i} ≤ n_{i}.

Corollary 57.22. Suppose the conditions of Proposition 57.21 hold and in addition the scheme W is reduced. Then [Y] · [Z] = ∑[V_{i}], i.e., all the m_{i} = 1.

If X is smooth, we write CH^{p}(X) for the group A^{p}(X, K_{p}) and call it the Chow group of codimension p classes of cycles on X. We now apply the results from §56 to this group. The graded group CH^{∗}(X) has the structure of a commutative associative ring with the identity 1_{X}. A morphism f : Y → X of smooth schemes induces a pull-back graded ring homomorphism f^{∗} : CH^{∗}(X) → CH^{∗}(Y).

Example 57.23. Let h ∈ CH^{1}(ℙ(V)) be the class of a hyperplane of the projective space ℙ(V), where V is a vector space of dimension d + 1, and l_{p} ∈ CH_{p}(ℙ(V)) as in Example 57.15. Then

h · l_{p} = l_{p−1}

for all p ∈ [1, d]. Indeed, h = [ℙ(U)] and l_{p} = [ℙ(V_{p})] where U and V_{p} are subspaces of V of dimensions n and p + 1 respectively. We can choose these subspaces so that the subspace V_{p−1} = U ∩V_{p} has dimension p. Then ℙ(U)∩ℙ(V_{p}) = ℙ(V_{p−1}) and we have equality by Corollary 57.22. It follows that CH^{p}(ℙ(V)) = ℤ · h^{p} for p ∈ [0, d]. In particular, the ring CH^{∗}(ℙ(V)) is generated by h with the one relation h^{d+1} = 0.

57.C. Cartier divisors and the Euler class. Let D be a Cartier divisor on a variety X of dimension d and L(D) its associated line bundle over X. Let [D] denote the associated divisor in Z_{d−1}(X). Recall that if D is a principal Cartier divisor given by a nonzero rational function f on X, then [D] = div(f).

Lemma 57.24. In the notation above, e(L(D))([X]) = [D] in CH_{d−1}(X).

Proof. Let p : L(D) → X and s : X → L(D) be the canonical morphism and the zero section respectively. Let X = ⋃ U_{i} be an open covering and f_{i} rational functions on U_{i} giving the Cartier divisor D. Let L(D) be the locally free sheaf of sections of L(D). The group of sections L(D)(U_{i}) consists of all rational functions f on X such that f ·f_{i} is regular on U_{i}. Thus we can view f_{i} as a section of the dual bundle L(D)^{∨} over U_{i}. The line bundle L(D) is the spectrum of the symmetric algebra

O_{X} ⊕ L(D)^{∨} · t ⊕ (L(D)^{∨})^{⊗2} · t^{2} ⊕ …

of the sheaf L(D)^{∨}. The rational functions (f_{i} · t)/f_{i} on p^{−1}(U_{i}) agree on intersections, so they give a well-defined rational function on L(D). We denote this function by t.

We claim that div(t) = s_{∗}([X]) − p^{∗}([D]). The statement is of a local nature, so we may assume that X is affine, say X = Spec(A) and D is a principal Cartier divisor given by a rational function f on X. We have L(D) = Spec(A[ft]) and by Proposition 49.23,

div(t) = div(ft) − div(p^{∗}f) = s_{∗}([X]) − p^{∗}(div(f)) = s_{∗}([X]) − p^{∗}([D])

proving the claim. By the claim, the classes s_{∗}([X]) and p^{∗}([D]) are equal in CH_{d}(L(D)). Hence, e(L(D))([X]) = (p^{∗})^{−1} ∘ s_{∗}([X]) = [D]. □

Example 57.25. Let C = Spec(S^{•}) be an integral cone with S^{•} a sheaf of graded O_{X}-algebras as in §104.A. Consider the cone C ⊕ 𝟙 = Spec(S^{•}[t]). The family of functions t/s with s ∈ S^{1} on the principal open subscheme D(s) of the projective bundle ℙ(C ⊕ 𝟙) gives rise to a Cartier divisor D on ℙ(C ⊕ 𝟙) with L(D) the canonical line bundle. The associated divisor D̃ coincides with ℙ(C). It follows from Lemma 57.24 that e(L(D))([ℙ(C ⊕ 𝟙)]) = [ℙ(C)].

Proposition 57.26. Let L and L′ be line bundles over a scheme X. Then e(L ⊗ L′) = e(L) + e(L′) on CH(X).

Proof. It suffices to prove that both sides of the equality coincide on the class [Z] of a closed subvariety Z in X. Denote by i : Z → X the closed embedding. Choose Cartier divisors D and D′ on Z so that L|_{Z} ≃ L(D) and L′|_{Z} ≃ L(D′). Then L|_{Z} ⊗ L′|_{Z} ≃ L(D + D′). By Proposition 53.3(1),

e(L ⊗ L′)([Z]) = i_{∗} ∘ e(L|_{Z} ⊗ L′|_{Z})([Z])
= i_{∗} ∘ e(L(D + D′))([Z])
= i_{∗}[{ \atop D + D′}]
= i_{∗}[D̃] + i_{∗}[D̃′]
= i_{∗} ∘ e(L(D)) + i_{∗} ∘ e(L(D′))
= e(L)([Z]) + e(L′)([Z]).  □

Corollary 57.27. For any line bundle L over X, we have e(L^{∨}) = −e(L).

58. Segre and Chern classes

In this section, we define Segre classes and consider their relations with Chern classes. The Segre class for a vector bundle is the inverse of the Chern class. The advantage of Segre classes is that they can be defined for arbitrary cones (not just for vector bundles like Chern classes). We follow the book [45] for the definition of Segre classes.

58.A. Segre classes. Let C = Spec(S^{•}) be a cone over X (cf. §104.A). Let q : ℙ(C ⊕ 𝟙) → X be the natural morphism and L the canonical line bundle over ℙ(C ⊕ 𝟙) (cf. §104). Let e(L)^{•} denote the total Euler class ∑_{k≥0} e(L)^{k} viewed as an operation on CH(ℙ(C ⊕ 𝟙)).

We define the Segre homomorphism

sg^{C} : CH(ℙ(C ⊕ 𝟙)) → CH(X) by
sg^{C} = q_{∗} ∘ e(L)^{•}.

The class Sg(C) := sg^{C}([ℙ(C ⊕ 𝟙)]) in CH(X) is known as the total Segre class of C.

Proposition 58.1. If C is a cone over X, then Sg(C ⊕ 𝟙) = Sg(C).

Proof. If [C] = ∑ m_{i}[C_{i}], where the C_{i} are the irreducible components of C, then

[ℙ(C ⊕ 𝟙^{k})] = ∑ m_{i}[ℙ(C_{i} ⊕ 𝟙^{k})]

for k ≥ 1. Therefore, we may assume that C is a variety. Let L and L′ be canonical line bundles over ℙ(C ⊕ 𝟙^{2}) and ℙ(C ⊕ 𝟙), respectively. We have L′ = i^{∗}(L), where i : ℙ(C ⊕ 𝟙) → ℙ(C ⊕ 𝟙^{2}) is the closed embedding. By Example 57.25, we have

e(L)([ℙ(C ⊕ 𝟙^{2})]) = [ℙ(C ⊕ 𝟙)],

Let q : ℙ(C ⊕ 𝟙^{2}) → X be the canonical morphism. It follows from Proposition 53.3(1) that

Sg(C ⊕ 𝟙) = q_{∗} ∘ e(L)^{•}([ℙ(C ⊕ 𝟙^{2})])
= q_{∗} ∘ e(L)^{•}(i_{∗}([ℙ(C ⊕ 𝟙)]))
= q_{∗}i_{∗} ∘ e(i^{∗}(L))^{•}([ℙ(C ⊕ 𝟙)])
= (q ∘ i)_{∗} ∘ e(L′)^{•}([ℙ(C ⊕ 𝟙)])
= Sg(C).  □

Proposition 58.2. Let C be a cone over a scheme X over F and i : Z → X a closed embedding. Let D be a closed subcone of the restriction of C on Z. Then the diagram

CH(ℙ(D ⊕ 𝟙)) ⟶^{sg^{D}} CH(Z)
j_{∗}↓  ↓^{i_{∗}}
CH(ℙ(C ⊕ 𝟙)) ⟶^{sg^{C}} CH(X)

is commutative, where j : ℙ(D ⊕ 𝟙) → ℙ(C ⊕ 𝟙) is the closed embedding. In particular, i_{∗}(Sg(D)) = sg^{C}(ℙ(D ⊕ 𝟙)).

Proof. The canonical line bundle L_{D} over ℙ(D ⊕ 𝟙) is the pull-back j^{∗}(L_{C}). It follows from the projection formula (cf. Proposition 53.3(1)) that

sg^{C} ∘j_{∗} = (q_{C})_{∗} ∘ e(L_{C})^{•} ∘ j_{∗}
= (q_{C})_{∗} ∘ j_{∗} ∘ e(j^{∗}(L_{C}))^{•}
= i_{∗} ∘ (q_{D})_{∗} ∘ e(L_{D})^{•}
= i_{∗} ∘ sg^{D} .  □

If C = E is a vector bundle over X, the projection q is a flat morphism of relative dimension r = rank E, and we define the total Segre operation s(E) on CH(X) by

s(E) : CH(X) → CH(X), s(E) = sg^{E} ∘ q^{∗} = q_{∗} ∘ e(L)^{•} ∘ q^{∗}.

In particular, Sg(E) = s(E)([X]).

For every k ∈ ℤ denote the degree k component of the operation s(E) by s_{k}(E), so it is the operation

s_{k}(E) : CH_{n}(X) → CH_{n−k}(X) given by
(58.3)  s_{k}(E) = q_{∗} ∘ e(L)^{k+r} ∘ q^{∗}.

Proposition 58.4. Let f : Y → X be a morphism of schemes over F and E a vector bundle over X. Set E′ = f^{∗}(E). Then

(1) If f is proper, then s(E) ∘ f_{∗} = f_{∗} ∘ s(E′).

(2) If f is flat, then f^{∗} ∘ s(E) = s(E′) ∘ f^{∗}.

Proof. Consider the fiber product diagram

ℙ(E′) ⟶^{h} ℙ(E)
q′↓  ↓^{q}
Y ⟶^{f} X

with flat morphisms q and q′ of constant relative dimension r −1 where r = rank E. Denote by e and e′ the Euler classes of the canonical line bundle L over ℙ(E) and L′ over ℙ(E′), respectively. Note that L′ = h^{∗}(L).

By Propositions 49.20 and 53.3, we have

s(E) ∘ f_{∗} = q_{∗} ∘ e(L)^{•} ∘ q^{∗} ∘ f_{∗}
= q_{∗} ∘ e(L)^{•} ∘ h_{∗} ∘ q′^{∗}
= q_{∗} ∘ h_{∗} ∘ e(L′)^{•} ∘ q′^{∗}
= f_{∗} ∘ q′_{∗} ∘ e(L′)^{•} ∘ q′^{∗}
= f_{∗} ∘ s(E′),

and

f^{∗} ∘ s(E) = f^{∗} ∘ q_{∗} ∘ e(L)^{•} ∘ q^{∗}
= q′_{∗} ∘ h^{∗} ∘ e(L)^{•} ∘ q^{∗}
= q′_{∗} ∘ e(L′)^{•} ∘ h^{∗} ∘ q^{∗}
= q′_{∗} ∘ e(L′)^{•} ∘ q′^{∗} ∘ f^{∗}
= s(E′) ∘ f^{∗}.  □

Proposition 58.5. Let E be a vector bundle over a scheme X over F. Then

s_{i}(E) = \{ \begin{matrix}0 & if i < 0, \\ id & if i = 0. \end{matrix}

Proof. Let α ∈ CH(X). We need to prove that s_{i}(E)(α) = 0 if i < 0 and s_{0}(E)(α) = α. We may assume that α = [Z], where Z ⊂ X is a closed subvariety. Let i : Z → X be the closed embedding. By Proposition 58.4(1), we have

s(E)(α) = s(E) ∘ i_{∗}([Z]) = i_{∗} ∘ s(E′)([Z]),

where E′ = i^{∗}(E). Hence it is sufficient to prove the statement for the vector bundle E′ over Z and the cycle [Z]. Therefore, we may assume that X is a variety of dimension d and α = [X] in CH_{d}(X). Since s_{i}(E)(α) ∈ CH_{d−i}(X), by dimension count, s_{i}(E)(α) = 0 if i < 0.

To prove the second identity, by Proposition 58.4(2), we may replace X by an open subscheme. Therefore, we can assume that E is a trivial vector bundle, i.e., ℙ(E) = X × ℙ^{r−1}. Applying Example 53.8 and Proposition 53.3(2) to the projection X × ℙ^{r−1} → ℙ^{r−1}, we have

s_{0}(E)([X]) = q_{∗} ∘ e(L)^{r−1} ∘ q^{∗}([X])
= q_{∗} ∘ e(L)^{r−1}([X] × ℙ^{r−1}) = q_{∗}([X] × ℙ^{0}) = [X].  □

Let E → X be a vector bundle of rank r. The restriction of Chern classes defined in §54 on Chow groups provides operations

c_{i}(E) : CH_{∗}(X) → CH_{∗−i}(X), α ↦ α_{i} = c_{i}(E)(α)

Example 58.6. In view of Examples 53.8 and 57.23, the class e(L) of the canonical line bundle L over ℙ^{d} acts on CH(ℙ^{d}) = ℤ[h]/(h^{d+1}) by multiplication by the class h of a hyperplane in ℙ^{d}.

By Example 104.20, the class of the tangent bundle of the projective space ℙ^{d} in K_{0}(ℙ^{d}) is equal to (d + 1)[L] − 1, hence c(T_{ℙ^{d}}) is multiplication by (1 + h)^{d+1}.

Example 58.7. For a vector bundle E, we have c_{i}(E^{∨}) = (−1)^{i}c_{i}(E). Indeed, by the Splitting Principle 53.13, we may assume that E has a filtration by subbundles with factors line bundles L_{1}, L_{2}, …, L_{r}. The dual bundle E^{∨} then has a filtration by subbundles with factors line bundles L_{1}^{∨}, L_{2}^{∨}, …, L_{r}^{∨}. As e(L_{k}^{∨}) = −e(L_{k}) by Corollary 57.27, it follows from Proposition 54.6 that

c_{i}(E^{∨}) = σ_{i}(e(L_{1}^{∨}), …, e(L_{r}^{∨})) = (−1)^{i}σ_{i}(e(L_{1}), …, e(L_{r})) = (−1)^{i}c_{i}(E)

where σ_{i} is the ith elementary symmetric function.

Let e and ẽ be the Euler classes of the tautological and the canonical line bundles over ℙ(E), respectively. By Corollary 57.27, we have ẽ = −e. Therefore, the formula (54.1) can be rewritten as

(58.8)  ∑_{i=0}^{r} ẽ^{r−i} ∘ q^{∗} ∘ c_{i}(E) = 0,

where q : ℙ(E) → X is the canonical morphism.

Proposition 58.9. Let E be a vector bundle over X. Then s(E) = c(E)^{−1}.

Proof. Applying q_{∗} ∘ ẽ^{k−1} to the equality (58.8) for the vector bundle E ⊕ 𝟙 of rank r + 1, we get for every k ≥ 1:

0 = ∑_{i≥0} q_{∗} ∘ ẽ^{r+k−i} ∘ q^{∗} ∘ c_{i}(E ⊕ 𝟙) = ∑_{i≥0} s_{k−i}(E) ∘ c_{i}(E ⊕ 𝟙)

in view of (58.3). By Corollary 54.9, we have c_{i}(E ⊕ 𝟙) = c_{i}(E). As s_{0}(E) = 1 and s_{i}(E) = 0 if i < 0 by Proposition 58.5, we have s(E) ∘ c(E) = 1. □

Proposition 58.10. Let E → X be a vector bundle and E′ ⊂ E a subbundle of corank r. Then

(58.11)  [ℙ(E′)] = ∑_{i=0}^{r} ẽ^{r−i} ∘ q^{∗} ∘ c_{i}(E/E′)([X])

in CH(ℙ(E)).

Proof. Applying (58.8) to the factor bundle E/E′, we have

∑_{i=0}^{r} e′ ^{r−i} ∘ q′^{∗} ∘ c_{i}(E/E′) = 0,

where q′ : ℙ(E/E′) → X is the canonical morphism and e′ is the Euler class of the canonical line bundle over ℙ(E/E′). Applying the pull-back homomorphism with respect to the canonical morphism ℙ(E) \ ℙ(E′) → ℙ(E/E′), we see that the restriction of the right hand side of the formula in (58.11) to ℙ(E) \ ℙ(E′) is trivial. By the localization property (cf. §52.D), the right hand side in (58.11) is equal to k[ℙ(E′)] for some k ∈ ℤ.

To determine k, we can replace X by an open subscheme of X and assume that E and E′ are trivial vector bundles of rank n and n − r, respectively. The right hand side in (58.11) is then equal to

ẽ^{r} ∘ q^{∗}([X]) = ẽ^{r}([ℙ^{n−1} × X]) = [ℙ^{n−r−1} × X] = [ℙ(E′)],

therefore k = 1. □

Proposition 58.12. Let E and E′ be vector bundles over schemes X and X′, respectively. Then

c(E × E′)(α × α′) = c(E)(α) × c(E′)(α′)

for any α ∈ CH(X) and α′ ∈ CH(X′).

Proof. Let p and p′ be the projections of X × X′ to X and X′, respectively. We claim that for any β ∈ CH(X) and β′ ∈ CH(X′), we have

(58.13)  c(p^{∗}(E))(β × β′) = c(E)(β) × β′,
(58.14)  c(p′^{∗}(E′))(β × α′) = β × c(E′)(β′).

To prove the claim, by Proposition 54.5, we may assume that β = [X] and β′ = [X′]. Then (58.13) and (58.14) follow from Proposition 54.5(2).

Since E × E′ = p^{∗}(E) ⊕ p′^{∗}(E′), by the Whitney Sum Formula 54.7 and by (58.13), (58.14), we have

c(E × E′)(α × α′) = c(p^{∗}(E) ⊕ p′^{∗}(E′))(α × α′)
= c(p^{∗}(E)) ∘ c(p′^{∗}(E′))(α × α′)
= c(p^{∗}(E))(α × c(E′)(α′))
= c(E)(α) × c(E′)(α′).  □

Proposition 58.15. Let E be a vector bundle over a smooth scheme X. Then c(E)(α) = c(E)([X]) · α for every α ∈ CH(X).

Proof. Consider the vector bundle E′ = E×X over X×X. Let d : X → X×X be the diagonal embedding. We have E = d^{∗}(E′). By Propositions 58.12 and 55.9,

c(E)(α) = c(d^{∗}(E′))(d^{⋆}([X] × α))
= d^{⋆}c(E × X)([X] × α)
= d^{⋆}(c(E)([X]) × α)
= c(E)([X]) · α.  □

Proposition 58.15 shows that for a vector bundle E over a smooth scheme X, the Chern class operation c(E) is multiplication by the class β = c(E)([X]). We shall sometimes write c(E) = β to mean that c(E) is multiplication by β.

Let f : Y → X be a morphism of schemes, i.e., Y is a scheme over X. Assume that X is a smooth variety. We shall see that CH(Y) has a natural structure of a module over the ring CH(X). Indeed, as we saw in §55.B, the morphism

i = (1_{Y} , f) : Y → Y × X

is a regular closed embedding of codimension dim X. For every α ∈ CH(Y) and β ∈ CH(X), we set

(58.16)  α · β = i^{⋆}(α × β).

Proposition 58.17. Let X be a smooth variety and Y a scheme over X. Then CH(Y) is a module over CH(X) under the product defined in (58.16). Let g : Y′ → Y be a proper (respectively flat) morphism of schemes over X. Then the homomorphism g_{∗} (respectively g^{∗}) is CH(X)-linear.

Proof. The composition of i and the projection p : Y ×X → Y is the identity on Y. It follows from Lemma 55.7 that α · [X] = i^{⋆}(α × [X]) = i^{⋆} ∘ p^{∗}(α) = 1_{Y}^{∗} (α) = α, i.e., the identity [X] of CH(X) acts on CH(Y) trivially.

Consider the fiber product diagram

Y ⟶^{i} Y × X
i↓  ↓^{h}
Y × X ⟶^{k} Y × X × X,

where k = 1_{Y} × d_{X} and h = i × 1_{X}. It follows from Corollary 55.4 that for any α ∈ CH(Y) and β, γ ∈ CH(X), we have

α · (β · γ) = i^{⋆}(α × (β · γ)) = i^{⋆}k^{⋆}(α × β × γ)

= i^{⋆}h^{⋆}(α × β × γ) = i^{⋆}((α · β) × γ) = (α · β) · γ.

Consider the fiber product diagram

Y′ ⟶^{i′} Y′ × X
g↓  ↓^{g×1_{X}}
Y ⟶^{i} Y × X.

Suppose first that the morphism g is proper. By Corollary 55.4,

g_{∗}(α′ · β) = g_{∗} ∘ i′^{⋆}(α′ × β) = i^{⋆}(g × 1_{X})_{∗}(α′ × β) = i′^{⋆}(g_{∗}(α′) × β) = g_{∗}(α′) · β

for all α′ ∈ CH(Y′) and β ∈ CH(X).

If g is flat, it follows from Proposition 55.5 that

g^{∗}(α · β) = g^{∗} ∘ i^{⋆}(α × β) = i′^{⋆} ∘ (g × 1_{X})^{∗}(α × β) = i′^{⋆}(g^{∗}(α) × β) = g^{∗}(α) · β

for all α ∈ CH(Y) and β ∈ CH(X). □

Proposition 58.18. Let f : Y → X be a morphism of schemes with X smooth and g : Y → Y′ a flat morphism. Suppose that for every point y′ ∈ Y′, the pull-back homomorphism CH(X) → CH(Y_{y′}) induced by the natural morphism of the fiber Y_{y′} to X is surjective. Then the homomorphism

h : CH(Y′) ⊗ CH(X) → CH(Y), α ⊗ β ↦ g^{∗}(α) · β

is surjective.

Proof. The proof is similar to the one given for Proposition 52.10. Obviously, we may assume that Y′ is reduced. Step 1: Y′ is a variety.

We induct on n = dim Y′. The case n = 0 is obvious. In general, let U′ ⊂ Y′ be a nonempty open subset and Z′ = Y′ \ U′ have the structure of a reduced scheme. Set U = g^{−1}(U′) and Z = g^{−1}(Z′). We have closed embeddings i : Z → Y, i′ : Z′ → Y′ and open immersions j : U → Y, j′ : U′ → Y′. By induction, the homomorphism h_{Z} is surjective in the diagram

CH(Z′) ⊗ CH(X) ⟶^{i′_{∗}⊗1} CH(Y′) ⊗ CH(X) ⟶^{j^{∗}⊗1} CH(U′) ⊗ CH(X) ⟶ 0
h_{Z} ↓  ^{h_{Y}} ↓  ^{h_{U}} ↓
CH(Z)  ⟶^{i_{∗}}  CH(Y)  ⟶^{j′^{∗}}  CH(U)  ⟶ 0

The diagram is commutative by Proposition 58.17.

Let y′ ∈ Y′ be the generic point. By Proposition 52.9, the colimit of the homomorphisms

(h_{U})^{∗} : CH(U′) ⊗ CH(X) → CH(U)

over all nonempty open subschemes U′ of Y′ is isomorphic to the pull-back homomorphism CH(X) → CH(Y_{y′}) which is surjective by assumption. Taking the colimits of all terms of the diagram, we conclude by the 5-lemma that h_{Y} is also surjective. Step 2: Y′ is an arbitrary scheme.

We induct on the number m of irreducible components of Y′. The case m = 1 is Step 1. Let Z′ be a (reduced) irreducible component of Y′ and let U′ = Y′ \ Z′. Consider the commutative diagram as in Step 1. By Step 1, the map h_{Z} is surjective. The map h_{U} is also surjective by the induction hypothesis. By the 5-lemma, h_{Y} is surjective. □

Proposition 58.19. Let C and C′ be cones over schemes X and X′, respectively. Then

Sg(C × C′) = Sg(C) × Sg(C′) ∈ CH(X × X′).

Proof. Set C̃ = C ⊕ 𝟙 and C̃′ = C′ ⊕ 𝟙. Let L and L′ be the tautological line bundles over ℙ(C̃) and ℙ(C̃′), respectively (cf. §104.D). We view L × L′ as a vector bundle over ℙ(C̃) × ℙ(C̃′). The canonical morphism L × L′ → C̃ × C̃′ induces a morphism

f : ℙ(L × L′) → ℙ(C̃ × C̃′).

If D is a cone, we write D^{∘} for the complement of the zero section in D. By §104.C, we have L^{∘} = C̃^{∘} and L′^{∘} = C̃^{′∘}. The open subsets C̃^{∘} × C̃^{′∘} in C̃ × C̃′ and L^{∘} ×L′^{∘} in L×L′ are dense. Hence f maps any irreducible component of ℙ(L×L′) birationally onto an irreducible component of ℙ(C̃ × C̃′). In particular,

f_{∗}([ℙ(L × L′)]) = [ℙ(C̃ × C̃′)].

Let L̃ be the canonical line bundle over ℙ(C̃ ×C̃′). Then f^{∗}(L̃) is the canonical line bundle over ℙ(L × L′). Let q : ℙ(C̃ × C̃′) → X × X′ be the natural morphism. By Proposition 58.1 and the Projection Formula 56.9, we have

Sg(C × C′) = Sg((C × C′) ⊕ 𝟙)
= q_{∗} ∘ e(L̃)^{•}([ℙ(C̃ × C̃′)])
= q_{∗} ∘ e(L̃)^{•}f_{∗}([ℙ(L × L′)])
= q_{∗} ∘ f_{∗} ∘ e(f^{∗}L̃)^{•}([ℙ(L × L′)]).

The normal bundle N of the closed embedding ℙ(C̃) × ℙ(C̃′) → L × L′, given by the zero section, coincides with L × L′. By definition of the Segre class and the Segre operation, we have

p_{∗} ∘ e(f^{∗}L̃)^{•}([ℙ(L × L′)]) = Sg(N) = s(N)([ℙ(C̃) × ℙ(C̃′)]),

where p : ℙ(L × L′) → ℙ(C̃) × ℙ(C̃′) is the natural morphism. By Propositions 58.12 and 58.9,

s(N)([ℙ(C̃) × ℙ(C̃′)]) = s(L)([ℙ(C̃)]) × s(L′)([ℙ(C̃′)]).

Let g : ℙ(C̃) → X and g′ : ℙ(C̃′) → X′ be the natural morphisms and set h = g × g′. By Proposition 50.4,

h_{∗} ∘ s(L)([ℙ(C̃)] × s(L′)([ℙ(C̃′)]))

= (g_{∗} ∘ s(L)([ℙ(C̃)])) × (g′_{∗} ∘ s(L′)([ℙ(C̃′)])) = Sg(C) × Sg(C′).

To finish the proof it is sufficient to notice that q ∘ f = h ∘ p and therefore q_{∗} ∘ f_{∗} = h_{∗} ∘ p_{∗}. □

Exercise 58.20. (Strong Splitting Principle) Let E be a vector bundle over X. Prove that there is a flat morphism f : Y → X such that the pull-back homomorphism f^{∗} : CH_{∗}(X) → CH_{∗}(Y) is injective and f^{∗}(E) is a direct sum of line bundles.

Exercise 58.21. Let E be a vector bundle of rank r. Prove that e(E) = c_{r}(E).

CHAPTER XI Steenrod Operations

In this chapter we develop Steenrod operations on Chow groups modulo 2. There are two reasons why we do not consider the operations modulo an arbitrary prime integer. First, this case is sufficient for our applications as the number 2 is the only “critical” prime for projective quadrics. Second, our approach does not immediately generalize to the case of an arbitrary prime integer.

Unfortunately, we need to assume that the characteristic of the base field is different from 2 in this chapter as we do not know how to define Steenrod operations in characteristic 2.

In this chapter, the word scheme means a quasi-projective scheme over a field F of characteristic not 2. We write Ch(X) for CH(X)/2 CH(X).

Let X be a scheme. Consider the homomorphism Z(X) → Ch(X) taking the class [Z] of a closed subvariety Z ⊂ X to j_{∗}(Sg(T_{Z})) modulo 2, where Sg is the total Segre class (cf. §58.A), T_{Z} is the tangent cone over Z (cf. Example 104.5) and j : Z → X is the closed embedding. We shall prove that this map factors through rational equivalence yielding the Steenrod operation modulo 2 of X (of homological type)

Sq^{X} : Ch(X) → Ch(X).

Thus we shall have

Sq^{X}([Z]) = j_{∗}(Sg(T_{Z}))

modulo 2. We shall see that the operation Sq^{X} commutes with the push-forward homomorphisms, so it can be viewed as a functor from the category of schemes to the category of abelian groups.

For a smooth scheme X, we can then define the Steenrod operation modulo 2 of X (of cohomological type) by the formula

Sq_{X} := c(T_{X}) ∘ Sq^{X} .

This formula can be viewed as a Riemann-Roch type relation between the operations Sq^{X} and Sq_{X}. We shall show that the operation Sq_{X} commutes with pull-back homomorphisms, so it can be viewed as a contravariant functor from the category of smooth schemes to the category of abelian groups.

In this chapter, we shall also prove the standard properties of the Steenrod operations.

Steenrod operations for motivic cohomology modulo a prime integer p of a smooth scheme X were originally constructed by Voevodsky in [137]. The reduced power operations (but not the Bockstein operation) restrict to the Chow groups of X. An “elementary” construction of the reduced power operations modulo p on Chow groups (requiring equivariant Chow groups) was given by Brosnan in [20].

59. Definition of the Steenrod operations

Consider a cyclic group G = {1, σ} of order 2. Let Y be a G-scheme such that p : Y → Y/G is a G-torsor over a field F (cf. §105.A). In particular, p is flat of relative dimension 0 by Proposition 105.5(1). For a point z ∈ Y/G, let Y_{z} be the fiber of p over z. By Example 105.6, we have Y_{z} = Spec(K), where K is an étale quadratic algebra over F (z). Then either K is a field (and the fiber Y_{z} has only one point y) or K = F (z) × F (z) (and the fiber has two points y_{1} and y_{2}). In any case, every point in Y_{z} is unramified (cf. Proposition 105.5(2)). It follows that for the pull-back homomorphism p^{∗} : Z(Y/G) → Z(Y), we have

p^{∗}([z]) = \{ \begin{matrix}[y] & if K is a field, \\ [y_{1}]+[y_{2}] & otherwise. \end{matrix}

Similarly, for a point y in the fiber Y_{z}, we have:

p_{∗}([y]) = \{ \begin{matrix}2[z] & if K is a field, \\ [z] & otherwise. \end{matrix}

In particular, p_{∗} ∘ p^{∗} is multiplication by 2.

We have σ(y) = y if K is a field and σ(y_{1}) = y_{2} otherwise. In particular,

(59.1)  p^{∗} ∘ p_{∗} = 1 + σ^{∗},

where σ^{∗} : Z(Y) → Z(Y) is the (pull-back) isomorphism induces by σ.

The cycles [y] and [y_{1}] + [y_{2}] generate the group Z(Y)^{G} of G-invariant cycles. We have proved

Proposition 59.2. Let p : Y → Y/G be a G-torsor. Then the pull-back homomorphism

p^{∗} : Z(Y/G) → Z(Y)^{G}

is an isomorphism.

Suppose char F ≠ 2. Let X be a scheme over F. The group G acts on X^{2}×𝔸^{1} = X × X × 𝔸^{1} by σ(x, x′, t) = (x′, x, −t). We have (X^{2} × 𝔸^{1})^{G} = Δ(X) × {0}, where Δ : X → X^{2} is the diagonal morphism (cf. §105.B). Set

(59.3)  U_{X} := (X^{2} × 𝔸^{1}) \ (Δ(X) × {0}).

The group G acts naturally on U_{X}. The morphism p : U_{X} → U_{X}/G is a G-torsor (cf. Example 105.4).

Let α ∈ Z(X) be a cycle. The cycle α^{2} × 𝔸^{1} := α × α × 𝔸^{1} in Z(X^{2} × 𝔸^{1}) is invariant under the above G-action as is the restriction of the cycle α^{2} × 𝔸^{1} on U_{X}. It follows from Proposition 59.2 that the pull-back homomorphism p^{∗} identifies Z(U_{X}/G) with Z(U_{X})^{G}. Let α_{G}^{2} ∈ Z(U_{X}/G) denote the cycle satisfying p^{∗}(α_{G}^{2}) = (α^{2} × 𝔸^{1})|_{U_{X}}.

We then have a map

(59.4)  Z(X) → Z(U_{X}/G), α ↦ α_{G}^{2}.

Lemma 59.5. If α and α′ are rationally equivalent cycles in Z(X), then α_{G}^{2} and α′_{G}^{2} are rationally equivalent cycles in Z(U_{X}/G).

Proof. As in §57.A, let Z(X; ℙ^{1}) denote the subgroup of Z(X × ℙ^{1}) generated by the classes of closed subvarieties in X × ℙ^{1} dominant over ℙ^{1}. Let W ⊂ X × ℙ^{1} and W′ ⊂ X′ × ℙ^{1} be two closed subvarieties dominant over ℙ^{1}. The projections W → ℙ^{1} and W′ → ℙ^{1} are flat and hence so is the fiber product W ×_{ℙ^{1}} W′ → ℙ^{1}.

Therefore, every irreducible component of W ×_{ℙ^{1}} W′ is dominant over ℙ^{1}, i.e., the cycle [W ×_{ℙ^{1}} W′] belongs to Z(X × X′; ℙ^{1}). By linearity, this construction extends to an external product over ℙ^{1} given by

Z(X; ℙ^{1}) × Z(X′; ℙ^{1}) → Z(X × X′; ℙ^{1}), (β, β′) ↦ β ×_{ℙ^{1}} β′.

By Proposition 57.12,

(59.6)  [(W ×_{ℙ^{1}} W′)(a)] = [W (a) × W′(a)] = [W (a)] × [W′(a)]

for any rational point a of ℙ^{1}. If X′ = X and β′ = β, write β̃^{2} for β ×_{ℙ^{1}} β.

By Proposition 57.6, there is a cycle β ∈ Z(X; ℙ^{1}) such that α = β(0) and α′ = β(∞). Consider the cycle β̃^{2} × [𝔸^{1}] ∈ Z(X^{2} × 𝔸^{1}; ℙ^{1}).

Let G act on X^{2} × 𝔸^{1} × ℙ^{1} by σ(x, x′, t, s) = (x′, x, −t, s). The cycle β̃^{2} × [𝔸^{1}] is G-invariant. Since U_{X} × ℙ^{1} is a G-torsor over (U_{X}/G) × ℙ^{1}, the restriction of the cycle β̃^{2} × [𝔸^{1}] on U_{X} × ℙ^{1} gives rise to a well-defined cycle

β̃_{G}^{2} ∈ Z(U_{X}/G; ℙ^{1})

satisfying

(59.7)  q^{∗}(β̃_{G}^{2}) = (β̃^{2} × [𝔸^{1}])|_{U_{X}×ℙ^{1}},

where q : U_{X} × ℙ^{1} → (U_{X}/G) × ℙ^{1} is the canonical morphism.

Let Z ⊂ (U_{X}/G) × ℙ^{1} be a closed subvariety dominant over ℙ^{1}. We have p^{−1}(Z(a)) = q^{−1}(Z)(a) for any rational point a of ℙ^{1}, where p : U_{X} → U_{X}/G is the canonical morphism. It follows from Proposition 57.8 that

(59.8)  p^{∗}(γ(a)) = (q^{∗}(γ))(a)

for every cycle γ ∈ Z(U_{X}/G; ℙ^{1}).

Let β = ∑ n_{i}[W_{i}]. Then applying (59.8) to γ = β̃_{G}^{2}, we see by (59.6) and (59.7) that

p^{∗}(β̃_{G}^{2}(a)) = (q^{∗}(β̃_{G}^{2}))(a) = (β̃^{2} × [𝔸^{1}])|_{U_{X}×ℙ^{1}}(a)
= ∑ n_{i}n_{j}[W_{i} ×_{ℙ^{1}} W_{j} × 𝔸^{1}]|_{U_{X}×ℙ^{1}}(a)
= ∑ n_{i}n_{j}[W_{i}(a) × W_{j}(a) × 𝔸^{1}]|_{U_{X}}
= p^{∗}(β(a)_{G}^{2})

in Z(U_{X}). It follows that β̃_{G}^{2}(a) = β(a)_{G}^{2} in Z(U_{X}/G) since p^{∗} is injective on cycles. In particular, β̃_{G}^{2}(0) = β(0)_{G}^{2} = α_{G}^{2} and β̃_{G}^{2}(∞) = β(∞)_{G}^{2} = α′_{G}^{2}, i.e., the cycles α_{G}^{2} and α′_{G}^{2} are rationally equivalent by Proposition 57.6. □

By Lemma 59.5, we have a well-defined map (but not a homomorphism!)

(59.9)  v_{X} : CH(X) → CH(U_{X}/G), [α] ↦ [α_{G}^{2}].

Consider the blowup B_{X} of X^{2} ×𝔸^{1} along Δ(X)×{0}. The exceptional divisor is the projective cone ℙ(T_{X} ⊕ 𝟙), where T_{X} is the tangent cone of X. The open complement B_{X} \ℙ(T_{X} ⊕𝟙) is naturally isomorphic to U_{X} and the group G = {1, σ} acts naturally on B_{X} (cf. §105.B).

By Proposition 105.7, the composition

i : ℙ(T_{X} ⊕ 𝟙) ↪ B_{X} → B_{X}/G

is a locally principal divisor with normal line bundle L^{⊗2} where L is the canonical line bundle over ℙ(T_{X} ⊕ 𝟙).

We define a map

u_{X} : Ch(U_{X}/G) → Ch(ℙ(T_{X} ⊕ 𝟙))

as follows: Let δ ∈ Ch(U_{X}/G). By the localization property (Proposition 57.9), there is β ∈ Ch(B_{X}/G) such that β|_{(U_{X}/G)} = δ. We set

u_{X}(δ) = i^{⋆}(β).

We claim that the result is independent of the choice of β. Indeed, if β′ ∈ Ch(B_{X}/G) is another element with β′|_{(U_{X}/G)} = δ, then by the localization, β′ = β + i_{∗}(γ) for some γ ∈ Ch(T_{X} ⊕ 𝟙). By Proposition 55.10, we have (i^{⋆} ∘ i_{∗})(γ) = e(L^{⊗2})(γ) = 2e(L)(γ) = 0 modulo 2, hence

i^{⋆}(β′) = i^{⋆}(β) + (i^{⋆} ∘ i_{∗})(γ) = i^{⋆}(β)

as needed.

Let q : B_{X} → B_{X}/G be the projection.

Lemma 59.10. The composition i^{⋆} ∘ q_{∗} : Ch(B_{X}) → Ch(ℙ(T_{X} ⊕ 𝟙)) is zero.

Proof. The scheme Y := q^{−1}(ℙ(T_{X} ⊕ 𝟙)) is a locally principal closed subscheme of B_{X}. The sheaf of ideals in O_{B_{X}} defining Y is the square of the sheaf of ideals of ℙ(T_{X} ⊕ 𝟙) as a subscheme of B_{X}. Let j : Y → B_{X} be the closed embedding and p : Y → ℙ(T_{X} ⊕ 𝟙) the natural morphism. By Corollary 55.4, we have i^{⋆} ∘ q_{∗} = p_{∗} ∘ j^{⋆}. It follows from Proposition 55.11 that j^{⋆} is trivial modulo 2. □

Proposition 59.11. For every scheme X, the map u_{X} is a homomorphism.

Proof. Let p : U_{X} → U_{X}/G be the projection. For any two cycles α = ∑ n_{i}[Z_{i}] and α′ = ∑ n′_{i}[Z_{i}] on X, we have

p^{∗}(α + α′)_{G}^{2} − p^{∗}(α_{G}^{2}) − p^{∗}(α′_{G}^{2}) = (1 + σ^{∗})(γ),

where

γ = ∑_{i<j} n_{i}n′_{j}[Z_{i} × Z_{j} × 𝔸^{1}]|_{U_{X}} ∈ Z(U_{X}).

Since p^{∗} ∘ p_{∗} = 1 + σ^{∗} by (59.1) and p^{∗} is injective on cycles by Proposition 59.2, we have

(α + α′)_{G}^{2} − α_{G}^{2} − α′_{G}^{2} = p_{∗}(γ).

Let β, β′, β′′ ∈ Ch(B_{X}/G) and δ ∈ Ch(U_{X}) be cycles restricting to α, α′, α + α′ and γ, respectively, satisfying

β′′ − β − β′ = q_{∗}(δ).

By Lemma 59.10,

u_{X}(α + α′) − u_{X}(α) − u_{X}(α′) = i^{⋆}(β′′) − i^{⋆}(β) − i^{⋆}(β′) = (i^{⋆} ∘ q_{∗})(δ) = 0. □

Definition 59.12. Let X be a scheme. We define the Steenrod operation of homological type as the composition

Sq^{X} : Ch(X) ⟶^{v_{X}} Ch(U_{X}/G) ⟶^{u_{X}} Ch(ℙ(T_{X} ⊕ 𝟙)) ⟶^{sg^{T_{X}}} Ch(X),

where sg^{T_{X}} is the Segre homomorphism defined in §58.A.

For every integer k we write

Sq_{k}^{X} : Ch_{∗}(X) → Ch_{∗−k}(X),

for the component of Sq^{X} decreasing dimension by k.

Proposition 59.13. Let Z be a closed subvariety of a scheme X. Then Sq^{X}([Z]) = j_{∗}(Sg(T_{Z})), where j : Z → X is the closed embedding and Sg is the total Segre class.

Proof. Let α = [Z] ∈ CH(X). We have v_{X}(α) = α_{G}^{2} = [U_{Z}/G]. Set β = [B_{Z}/G] ∈ CH(B_{Z}/G). By Proposition 55.6, we have i_{Z}^{⋆}(β) = [ℙ(T_{Z} ⊕ 𝟙)], where i_{Z} : ℙ(T_{Z} ⊕ 𝟙) → B_{Z}/G is the closed embedding.

Consider the diagram

Ch(B_{Z}/G) ⟶^{i_{Z}^{⋆}} Ch(ℙ(T_{Z} ⊕ 𝟙)) ⟶^{sg^{T_{Z}}} Ch(Z)
k_{∗}↓  ↓  ^{j_{∗}}↓
Ch(B_{X}/G) ⟶^{i_{X}^{⋆}} Ch(ℙ(T_{X} ⊕ 𝟙)) ⟶^{sg^{T_{X}}} Ch(X)

with vertical maps the push-forward homomorphisms. The diagram is commutative by Corollary 55.4 and Proposition 58.2. The commutativity yields

Sq^{X}([Z]) = (sg^{T_{X}} ∘i_{X}^{⋆})(k_{∗}(β))
= (j_{∗} ∘ sg^{T_{Z}} ∘i_{Z}^{⋆})(β)
= (j_{∗} ∘ sg^{T_{Z}})([ℙ(T_{Z} ⊕ 𝟙)])
= j_{∗}(Sg(T_{Z})).  □

Remark 59.14. The maps v_{X}, u_{X} and sg^{T_{X}} commute with arbitrary field extensions, hence so do Steenrod operations. More precisely, if L/F is a field extension, then the diagram

Ch(X) ⟶^{Sq^{X}} Ch(X)
↓  ↓
Ch(X_{L}) ⟶^{Sq^{X_{L}}} Ch(X_{L})

commutes.

60. Properties of the Steenrod operations

In this section, we establish the standard properties of Steenrod operations of homological type.

60.A. Formula for a smooth cycle. Let Z be a smooth closed subvariety of a scheme X. By Proposition 58.9, the total Segre class Sg(T_{Z}) coincides with s(T_{Z})([Z]) = c(T_{Z})^{−1}([Z]) = c(−T_{Z})([Z]), where c is the total Chern class. Hence by Proposition 59.13,

(60.1)  Sq^{X}([Z]) = j_{∗} ∘ c(−T_{Z})([Z]),

where j : Z → X is the closed embedding.

60.B. External products.

Theorem 60.2. Let X and Y be two schemes over a field F of characteristic not 2. Then Sq^{X×Y} (α × β) = Sq^{X}(α) × Sq^{Y} (β) for any α ∈ Ch(X) and β ∈ Ch(Y). Equivalently,

Sq_{n}^{X×Y} (α × β) = ∑_{k+m=n} Sq_{k}^{X}(α) × Sq_{m}^{Y}(β)

for all n.

Proof. We may assume that α = [V] and β = [W] where V and W are closed subvarieties of X and Y, respectively. Let i : V → X and j : W → Y be the closed embeddings. By Propositions 50.4, 58.19, and Corollary 104.8,

Sq^{X×Y} (α × β) = (i × j)_{∗} ∘ Sg(T_{V ×W})
= (i_{∗} × j_{∗}) ∘ Sg(T_{V} × T_{W})
= (i_{∗} × j_{∗}) ∘ (Sg(T_{V}) × Sg(T_{W}))
= i_{∗} ∘ Sg(T_{V}) × j_{∗} ∘ Sg(T_{W})
= Sq^{X}(α) × Sq^{Y} (β).  □

60.C. Functoriality of Sq^{X}.

Lemma 60.3. Let i : Y → X be a closed embedding. Then i_{∗} ∘ Sq^{Y} = Sq^{X} ∘i_{∗}.

Proof. Let Z ⊂ Y be a closed subscheme and let j : Z → Y be the closed embedding. By Proposition 59.13, we have

i_{∗} ∘ Sq^{Y} ([Z]) = i_{∗} ∘ j_{∗} ∘ Sg(T_{Z}) = (ij)_{∗} ∘ Sg(T_{Z}) = Sq^{X}(i_{∗}[Z]).  □

Lemma 60.4. Let p : ℙ^{r} ×X → X be the projection. Then p_{∗} ∘Sq^{ℙ^{r}×X} = Sq^{X} ∘p_{∗}.

Proof. The group CH(ℙ^{r} × X) is generated by cycles α = [ℙ^{k} × Z] for all closed subvarieties Z ⊂ X and k ≤ r by Theorem 57.14. It follows from Lemma 60.3 that we may assume Z = X and k = r. The statement is obvious if r = 0, so we may assume that r > 0. Since p_{∗}(α) = 0, we need to prove that p_{∗} Sq^{ℙ^{r}×X}(α) = 0.

By Theorem 60.2, we have

Sq^{ℙ^{r}×X}(α) = Sq^{ℙ^{r}}([ℙ^{r}]) × Sq^{X}([X]).

It follows from Example 58.6 and (60.1) that

Sq^{ℙ^{r}}([ℙ^{r}]) = c(T_{ℙ^{r}})^{−1}([ℙ^{r}]) = (1 + h)^{−r−1},

where h = c_{1}(L) is the class of a hyperplane in ℙ^{r}. By Proposition 50.4,

p_{∗}(Sq^{ℙ^{r}×X}(α)) = deg(1 + h)^{−r−1} · Sq^{X}([X]).

We have

deg(1 + h)^{−r−1} = \binom{−r − 1}{r} = (−1)^{r}\binom{2r}{r}

and the latter binomial coefficient is even if r > 0. □

Theorem 60.5. Let f : Y → X be a projective morphism. Then the diagram

Ch(Y) ⟶^{Sq^{Y}} Ch(Y)
f_{∗}↓  ^{f_{∗}}↓
Ch(X) ⟶^{Sq^{X}} Ch(X)

is commutative.

Proof. The projective morphism f factors as the composition of a closed embedding Y → ℙ^{r} × X and the projection ℙ^{r} × X → X, so the statement follows from Lemmas 60.3 and 60.4. □

Theorem 60.6. Sq_{k}^{X} = 0 if k < 0 and Sq_{0}^{X} is the identity.

Proof. First suppose that X is a variety of dimension d. By dimension count, the class Sq_{k}^{X}([X]) = Sg_{d−k}(T_{X}) is trivial if k < 0. To compute Sq_{0}^{X}([X]), we can extend the base field to a perfect one and replace X by a smooth open subscheme. Then by (60.1),

Sq_{0}^{X}([X]) = c_{0}(−T_{X})([X]) = [X],

i.e., Sq_{0}^{X} is the identity on Ch_{d}(X).

In general, let Z ⊂ X be a closed subvariety and let j : Z → X be the closed embedding. Then by Lemma 60.3 and the first part of the proof, the class Sq_{k}^{X}([Z]) = j_{∗}(Sq_{k}^{Z}([Z])) is trivial for k < 0 and is equal to [Z] ∈ Ch(X) if k = 0. □

61. Steenrod operations for smooth schemes

In this section, we define Steenrod operations of cohomological type and prove their basic properties.

Lemma 61.1. Let f : Y → X be a regular closed embedding of schemes of codimension r and g : U_{Y} /G → U_{X}/G the closed embedding induced by f. Then g is a regular closed embedding of codimension 2r and the following diagram is commutative:

CH(X) ⟶^{v_{X}} CH(U_{X}/G)
f^{⋆}↓  ↓_{g^{⋆}}
CH(Y) ⟶^{v_{Y}} CH(U_{Y} /G)

Proof. The closed embedding U_{Y} → U_{X} is regular of codimension 2r and the morphism U_{X} → U_{X}/G is faithfully flat. Hence g is also a regular closed embedding by Proposition 104.11. Let p : N → Y be the normal bundle of f. The Gysin homomorphism f^{⋆} is the composition of the deformation homomorphism σ_{f} : CH(X) → CH(N) and the inverse to the pull-back isomorphism p_{f}^{∗} : CH(Y) → CH(N) (cf. §55.A).

The normal bundle N_{h} of the closed embedding h : U_{Y} → U_{X} is the restriction of the vector bundle N^{2} × 𝔸^{1} on U_{Y}.

Consider the diagram (61.2)

Z(X) ⟶ Z(X^{2} × 𝔸^{1})^{G} ⟶ Z(U_{X})^{G}  Z(U_{X}/G)
↓^{σ_{f}}  ↓^{σ_{f^{2}×1}}  ↓^{σ_{h}}  ↓^{σ_{g}}
Z(N) ⟶ Z(N^{2} × 𝔸^{1})^{G} ⟶ Z(N_{h})^{G}  Z(N_{h}/G)
↑  ↑  ↑  ↑
Z(Y) ⟶ Z(Y^{2} × 𝔸^{1})^{G} ⟶ Z(U_{Y})^{G}  Z(U_{Y} /G)

where the first homomorphism in every row takes a cycle α to α^{2} × [𝔸^{1}], the other unmarked maps are pull-back homomorphisms with respect to flat morphisms and the equalities follow from Proposition 59.2.

The deformation homomorphism is defined by σ_{f}(∑ n_{i}[Z_{i}]) = [C_{k_{i}}], where k_{i} : Y ∩ Z_{i} → Z_{i} is the restriction of f by Proposition 52.7, so the commutativity of the upper left square follows from the equality of cycles [C_{k_{i}} × C_{k_{j}}] = [C_{k_{i}×k_{j}}] (cf. Proposition 104.7). The two other top squares are commutative by Proposition 51.5. The commutativity of the left bottom square follows from Propositions 57.8 and 57.12. The two other squares are commutative by Proposition 49.18.

The normal bundle N_{h} is an open subscheme of U_{N} and of N^{2} × 𝔸^{1}. Let j : N_{h} → U_{N} and l : N_{h}/G → U_{N}/G be the open embeddings. The following diagram of the pull-back homomorphisms

Z(N) ⟶ Z(N^{2} × 𝔸^{1})^{G} ⟶ Z(U_{N})^{G}  Z(U_{N}/G)
∥  ∥  ^{j^{∗}}↓  ↓_{l^{∗}}
Z(N) ⟶ Z(N^{2} × 𝔸^{1})^{G} ⟶ Z(N_{h})^{G}  Z(N_{h}/G)

is commutative by Proposition 49.18. It follows from Lemma 59.5 that the composition in the top row factors through the rational equivalence, hence so does the composition in the bottom row and then in the middle row of the diagram (61.2). Therefore the diagram (61.2) yields a commutative diagram:

\xymatrix{CH(X) \ar[d]_{σ_{f}} \ar[r]^{v_{X}} & CH(U_{X}/G) \ar[dr]^{σ_{g}} \\ CH(N) \ar[r] & CH(U_{N}/G) \ar[r]^{l_{∗}} & CH(N_{h}/G) \\ CH(Y) \ar[u]^{p_{f}^{∗}}_{≀} \ar[r]^{v_{Y}} & CH(U_{Y} /G) \ar[ur]_{p_{g}^{∗}}^{∼}}

The lemma follows from the commutativity of this diagram. □

Let f : Y → X be a closed embedding of smooth schemes with the normal bundle N → Y. Consider the diagram

ℙ(T_{Y} ⊕ 𝟙) ⟶^{j} ℙ(T_{X} ⊕ 𝟙)
p↓  ^{q}↓
Y  ⟶^{f}  X.

Lemma 61.3. We have c(N) ∘ f^{∗} ∘ sg^{T_{X}} = sg^{T_{Y}} ∘j^{∗}.

Proof. By the Projective Bundle Theorem 57.14, the group CH(ℙ(T_{X} ⊕ 𝟙)) is generated by the elements β = e(L_{X})^{k}(q^{∗}(α)) with k ≥ 0 and α ∈ CH(X). We have

(61.4)  e(L_{X})^{•}(β) = e(L_{X})^{•}(q^{∗}(α)).

Since j^{∗}(L_{X}) = L_{Y} and j^{∗} ∘ q^{∗} = p^{∗} ∘ f^{∗}, we have j^{∗}β = e(L_{T_{Y}})^{k}(p^{∗}(f^{∗}(α))) by Proposition 53.3(2), and therefore,

(61.5)  e(L_{Y})^{•}(j^{∗}(β)) = e(L_{Y})^{•} ∘ p^{∗}(f^{∗}(α)).

By Proposition 104.16, c(N)∘s(f^{∗}(T_{X})) = c(N)∘c(f^{∗}(T_{X}))^{−1} = c(T_{Y})^{−1} = s(T_{Y}). It follows from (61.4), (61.5), Propositions 54.7 and 58.4(2) that

c(N) ∘ f^{∗} ∘ sg^{T_{X}} (β) = c(N) ∘ f^{∗} ∘ q^{∗} ∘ e(L_{X})^{•}(β)
= c(N) ∘ f^{∗} ∘ q^{∗} ∘ e(L_{X})^{•}(q^{∗}(α))
= c(N) ∘ f^{∗} ∘ s(T_{X})(α)
= c(N) ∘ s(f^{∗}T_{X})(f^{∗}(α))
= s(T_{Y})(f^{∗}(α))
= p^{∗} ∘ e(L_{Y})^{•} ∘ p^{∗}(f^{∗}(α))
= p^{∗} ∘ e(L_{Y})^{•}(j^{∗}(β))
= sg^{T_{Y}} ∘j^{∗}(β).  □

Proposition 61.6. Let f : Y → X be a closed embedding of smooth schemes with the normal bundle N. Then c(N) ∘ f^{∗} ∘ Sq^{X} = Sq^{Y} ∘f^{∗}.

Proof. By Proposition 105.8, the schemes B_{Y} /G and B_{X}/G are smooth. Let

j : ℙ(T_{Y} ⊕ 𝟙) → ℙ(T_{X} ⊕ 𝟙) and h : B_{Y} /G → B_{X}/G

be the closed embeddings induced by f. Let α ∈ Ch(X). Choose β ∈ Ch(B_{X}/G) satisfying β|_{(U_{X}/G)} = α_{G}^{2} (cf. (59.4)). It follows from Proposition 55.19 and Lemma 61.1 that

(h^{∗}(β))|_{U_{Y} /G} = (f^{∗}(α))_{G}^{2}.

By Proposition 55.19 and Lemma 61.3,

c(N) ∘ f^{∗} ∘ Sq^{X}(α) = c(N) ∘ f^{∗} ∘ sg^{T_{X}} ∘i_{X}^{⋆}(β)
= sg^{T_{Y}} ∘j^{∗} ∘ i_{X}^{⋆}(β)
= sg^{T_{Y}} ∘i_{Y}^{⋆} ∘ h^{∗}(β)
= Sq^{Y} ∘f^{∗}(α).  □

Definition 61.7. Let X be a smooth scheme. We define the Steenrod operations of cohomological type by the formula

Sq_{X} = c(T_{X}) ∘ Sq^{X} .

We write Sq_{X}^{k} for kth homogeneous part of Sq_{X}. Thus Sq_{X}^{k} is an operation

Sq_{X}^{k} : Ch^{∗}(X) → Ch^{∗+k}(X).

Proposition 61.8 (Wu Formula). Let Z be a smooth closed subscheme of a smooth scheme X. Then Sq_{X}([Z]) = j_{∗} ∘ c(N)([Z]), where N is the normal bundle of the closed embedding j : Z → X.

Proof. By Proposition 54.5 and (60.1),

Sq_{X}([Z]) = c(T_{X}) ∘ Sq^{X}([Z])
= c(T_{X}) ∘ j_{∗} ∘ c(−T_{Z})([Z])
= j_{∗} ∘ c(i^{∗}(T_{X})) ∘ c(−T_{Z})([Z])
= j_{∗} ∘ c(N)([Z])

since c(T_{Z}) ∘ c(N) = c(j^{∗}(T_{X})). □

Theorem 61.9. Let f : Y → X be a morphism of smooth schemes. Then the diagram

Ch(X) ⟶^{Sq_{X}} Ch(X)
f^{∗}↓  ↓^{f^{∗}}
Ch(Y) ⟶^{Sq_{Y}} Ch(Y)

is commutative.

Proof. Suppose first that f is a closed embedding with normal bundle N. It follows from Propositions 54.5(2) and 61.6 that

f^{∗} ∘ Sq_{X} = f^{∗} ∘ c(T_{X}) ∘ Sq^{X}
= c(f^{∗}(T_{X})) ∘ f^{∗} ∘ Sq^{X}
= c(T_{Y}) ∘ c(N) ∘ f^{∗} ∘ Sq^{X}
= c(T_{Y}) ∘ Sq^{Y} ∘f^{∗}
= Sq_{Y} ∘f^{∗}.

Second, consider the case of the projection g : Y × X → X. Let Z ⊂ X be a closed subvariety. By (60.1), Propositions 58.12, 57.12, Corollary 104.8, and Theorem 60.2, we have g^{∗}([Z]) = [Y × Z] = [Y] × [Z] and

Sq_{Y ×X}(g^{∗}([Z])) = c(T_{Y ×X}) ∘ Sq^{Y ×X}([Y × Z])
= (c(T_{Y}) × c(T_{X}))(Sq^{Y} ([Y]) × Sq^{X}([Z]))
= (c(T_{Y}) ∘ Sq^{Y} ([Y])) × (c(T_{X}) ∘ Sq^{X}([Z]))
= [Y] × Sq_{X}([Z])
= g^{∗}(Sq_{X}([Z])).

In the general case, write f = g ∘ h where h = (id_{X}, f) : Y → Y × X is the closed embedding and g : Y × X → X is the projection. Then by the above,

f^{∗} ∘ Sq_{X} = h^{∗} ∘ g^{∗} ∘ Sq_{X} = h^{∗} ∘ Sq_{Y ×X} ∘g^{∗} = Sq_{Y} ∘h^{∗} ∘ g^{∗} = Sq_{Y} ∘f^{∗}. □

Proposition 61.10. Let f : Y → X be a smooth projective morphism of smooth schemes. Then

Sq_{X} ∘f_{∗} = f_{∗} ∘ c(−T_{f}) ∘ Sq_{Y} ,

where T_{f} is the relative tangent bundle of f.

Proof. It follows from the exactness of the sequence

0 → T_{f} → T_{Y} → f^{∗}(T_{X}) → 0

that c(T_{Y}) = c(T_{f}) ∘ c(f^{∗}T_{X}). By Proposition 54.5(1) and Theorem 60.5,

Sq_{X} ∘f_{∗} = c(T_{X}) ∘ Sq^{X} ∘f_{∗}
= c(T_{X}) ∘ f_{∗} ∘ Sq^{Y}
= c(T_{X}) ∘ f_{∗} ∘ c(−T_{Y}) ∘ Sq_{Y}
= f_{∗} ∘ c(f^{∗}T_{X}) ∘ c(−T_{Y}) ∘ Sq_{Y}
= f_{∗} ∘ c(−T_{f}) ∘ Sq_{Y} .  □

Let X be a smooth variety of dimension d and let Z ⊂ X be a closed subvariety. Consider the closed embedding j : ℙ(T_{Z}⊕𝟙) → ℙ(T_{X}⊕𝟙). By the Projective Bundle Theorem 57.14 applied to the vector bundle T_{X} ⊕ 𝟙 over X of rank d + 1, there are unique elements α_{0}, α_{1}, …, α_{d} ∈ Ch(X) such that

j_{∗}([ℙ(T_{Z} ⊕ 𝟙)]) = ∑_{k=0}^{d} e(L)^{k}(q^{∗}(α_{k}))

in Ch(ℙ(T_{X} ⊕ 𝟙)), where L is the canonical line bundle over ℙ(T_{X} ⊕ 𝟙) and q : ℙ(T_{X} ⊕ 𝟙) → X is the natural morphism. We set α := α_{0} + α_{1} + ⋯ + α_{d} ∈ Ch(X).

Lemma 61.11. Sq^{X}([Z]) = s(T_{X})(α).

Proof. Let p : ℙ(T_{Z} ⊕ 𝟙) → Z be the projection and i : Z → X the closed embedding, so that i ∘ p = q ∘ j. The canonical line bundle L′ over ℙ(T_{Z} ⊕ 𝟙) coincides with j^{∗}(L) and by Proposition 53.3,

Sq^{X}([Z]) = i_{∗} Sg(T_{Z})
= i_{∗} ∘ p_{∗} ∘ e(L′)^{•}([ℙ(T_{Z} ⊕ 𝟙)])
= q_{∗} ∘ j_{∗} ∘ e(j^{∗}(L))^{•}([ℙ(T_{Z} ⊕ 𝟙)])
= q_{∗} ∘ e(L)^{•} ∘ j_{∗}([ℙ(T_{Z} ⊕ 𝟙)])
= q_{∗} ∘ e(L)^{•} ∘ ∑_{k=0}^{d} e(L)^{k}(q^{∗}(α_{k}))
= q_{∗} ∘ e(L)^{•} ∘ q^{∗}(α)
= s(T_{X})(α).  □

Corollary 61.12. Sq_{X}([Z]) = α in Ch(X).

Proof. By Lemma 61.11 and Proposition 58.9,

Sq_{X}([Z]) = c(T_{X})(Sq^{X}([Z])) = c(T_{X})s(T_{X})(α) = α.  □

Theorem 61.13. Let X be a smooth scheme. Then for any β ∈ Ch^{k}(X),

Sq_{X}^{r}(β) = \begin{cases}β & if r = 0, \\ β^{2} & if r = k, \\ 0 & if r < 0 or r > k. \end{cases}

Proof. By definition and Theorem 60.6, we have Sq_{X}^{k} = 0 if k < 0 and Sq_{X}^{0} is the identity operation.

We may assume that X is a variety and β = [Z] where Z ⊂ X is a closed subvariety of codimension k. Since α_{i} ∈ Ch^{2k−i}(X), we have Sq_{X}^{r}(β) = α_{k−r} by Corollary 61.12. Therefore, Sq_{X}^{r}(β) = 0 if r > k.

As Sq_{X}^{k}(β) = α_{0}, it remains to prove that β^{2} = α_{0}. Consider the diagonal embedding d : X → X^{2} and the closed embedding h : T_{Z} → T_{X}. By the definition of the product in Ch(X) and Proposition 52.7,

p^{∗}(β^{2}) = p^{∗} ∘ d_{X}^{∗}([Z^{2}]) = σ_{d}([Z^{2}]) = h_{∗}([T_{Z}]) ∈ Ch(T_{X}),

where p : T_{X} → X is the canonical morphism. Let j : T_{X} → ℙ(T_{X} ⊕ 𝟙) be the open embedding. Since the pull-back j^{∗}(L) of the canonical line bundle L over ℙ(T_{X} ⊕𝟙) is a trivial line bundle over T_{X}, we have

j^{∗} ∘ e(L)^{s}(q^{∗}(α)) = e(j^{∗}(L))^{s}(j^{∗} ∘ q^{∗}(α)) = \begin{cases}p^{∗}(α) & if s = 0, \\ 0 & if s > 0 \end{cases}

for every α ∈ Ch(X). Hence

p^{∗}(β^{2}) = [T_{Z}] = j^{∗}([ℙ(T_{Z} ⊕ 𝟙)]) = p^{∗}(α_{0}),

therefore, β^{2} = α_{0} since p^{∗} is an isomorphism. □

Theorem 61.14. Let X and Y be two smooth schemes. Then

Sq_{X×Y} = Sq_{X} × Sq_{Y} .

Proof. By Corollary 104.8, we have T_{X×Y} = T_{X} × T_{Y}. It follows from Theorem 60.2 and Proposition 58.12 that

Sq_{X×Y} = c(T_{X×Y}) ∘ Sq^{X×Y}
= (c(T_{X}) ∘ Sq^{X}) × (c(T_{Y}) ∘ Sq^{Y} )
= Sq_{X} × Sq_{Y} .  □

Corollary 61.15 (Cartan Formula). Let X be a smooth scheme. Then Sq_{X}(α · β) = Sq_{X}(α) · Sq_{X}(β) for all α, β ∈ Ch(X). Equivalently,

Sq_{X}^{n}(α · β) = ∑_{k+m=n} Sq_{X}^{k}(α) · Sq_{X}^{m}(β)

for all n.

Proof. Let i : X → X × X be the diagonal embedding. Then by Theorems 61.9 and 61.14,

Sq_{X}(α · β) = Sq_{X}(i^{∗}(α × β))
= i^{∗} Sq_{X×Y} (α × β)
= i^{∗}(Sq_{X}(α) × Sq_{X}(β))
= Sq_{X}(α) · Sq_{X}(β).  □

Example 61.16. Let X = ℙ^{d} be projective space and h ∈ Ch^{1}(X) the class of a hyperplane. By Theorem 61.13, we have Sq_{X}(h) = h + h^{2} = h(1 + h). It follows from Corollary 61.15 that

Sq_{X}(h^{i}) = h^{i}(1 + h)^{i}, Sq_{X}^{r}(h^{i}) = \binom{i}{r}h^{i+r}.

By Example 104.20, the class of the tangent bundle T_{X} is equal to (d + 1)[L] − 1, where L is the canonical line bundle over X. Hence

c(T_{X}) = c(L)^{d+1} = (1 + h)^{d+1}

and

Sq^{X}(h^{i}) = c(T_{X})^{−1} ∘ Sq_{X}(h^{i}) = h^{i}(1 + h)^{i−d−1}.

CHAPTER XII Category of Chow Motives

In this chapter we study Chow motives. The notion of a Chow motive is due to Grothendieck. Many (co)homology theories defined on the category Sm(F) of smooth complete schemes, such as Chow groups and more generally K-(co)homology groups take values in the category of abelian groups. But the category Sm(F) itself does not have the structure of an additive category as we cannot add morphisms of schemes.

In this chapter, for an arbitrary commutative ring Λ, we shall construct the additive categories of correspondences CR(F, Λ), CR_{∗}(F, Λ) and motives CM(F, Λ), CM_{∗}(F, Λ) together with functors

Sm(F) ⟶ CR(F, Λ) ⟶ CM(F, Λ)
↓  ↓
CR_{∗}(F, Λ) ⟶ CM_{∗}(F, Λ)

so that the theories with values in the category of abelian groups mentioned above factor through them. All of these the new categories do have the additional structure of additive category. This makes them easier to work with than the category Sm(F). Applications of these categories can be found in Chapter XVII later in this book.

Some classical theorems also have motivic analogs. For example, the Projective Bundle Theorem 53.10 has such an analog (cf. Theorem 63.10) below. We shall see that the motive of a projective bundle splits into a direct sum of certain motives already in the category of correspondences CR(F, Λ). From this the classical Projective Bundle Theorem 53.10 can be obtained by applying an appropriate functor to the decomposition in CR(F, Λ).

In this chapter, scheme means a separated scheme of finite type over a field.

62. Correspondences

Definition 62.1. Let X and Y be two schemes over F. A correspondence between X and Y is an element of CH(X × Y).

For example, the graph of a morphism between X and Y is a correspondence. In this section we study functorial properties of correspondences.

For a scheme Y over F, we have two canonical morphisms: the structure morphism p_{Y} : Y → Spec(F) and the diagonal closed embedding d_{Y} : Y → Y × Y. If Y is complete, the map p_{Y} is proper, and if Y is smooth, the closed embedding d_{Y} is regular.

Let X, Y and Z be schemes over F with Y complete and smooth. We consider morphisms

Xp_{Y}^{Z} := 1_{X} × p_{Y} × 1_{Z} : X × Y × Z → X × Z

and

Xd_{Y}^{Z} := 1_{X} × d_{Y} × 1_{Z} : X × Y × Z → X × Y × Y × Z.

If X = Spec(F), we shall simply write p_{Y}^{Z} and d_{Y}^{Z}.

Define a bilinear pairing of K-homology groups (cf. §52)

A_{∗}(Y × Z, K_{∗}) × A_{∗}(X × Y, K_{∗}) → A_{∗}(X × Z, K_{∗})

by

(62.2)  (β, α) ↦ β ∘ α := (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆}(α × β).

For an element α ∈ A_{∗}(X × Y, K_{∗}) we write α^{t} for its image in A_{∗}(Y × X, K_{∗}) under the exchange isomorphism X × Y ≃ Y × X. The element α^{t} is called the transpose of α. By the definition of the pairing,

(β ∘ α)^{t} = α^{t} ∘ β^{t}.

Proposition 62.3. The pairing (62.2) is associative. More precisely, for any four schemes X, Y, Z, T over F with Y and Z smooth and complete and any α ∈ A_{∗}(X × Y, K_{∗}), β ∈ A_{∗}(Y × Z, K_{∗}), and γ ∈ A_{∗}(Z × T, K_{∗}), we have

(γ ∘ β) ∘ α = (^{X}p_{Y ×Z}^{T})_{∗} ∘ (^{X}d_{Y ×Z}^{T})^{⋆}(α × β × γ) = γ ∘ (β ∘ α).

Proof. We prove the first equality. It follows from Corollary 55.4 that

(^{X}d_{Y}^{T} )^{⋆} ∘ (^{X×Y ×Y}p_{Z}^{T})_{∗} = (^{X×Y}p_{Z}^{T})_{∗} ∘ (^{X}d_{Y}^{Z×T} )^{⋆}.

By Propositions 50.4, 50.5, and 55.1, we have

(γ ∘ β) ∘ α = (^{X}p_{Y}^{T} )_{∗} ∘ (^{X}d_{Y}^{T} )^{⋆}(α × (^{Y}p_{Z}^{T})_{∗}(^{Y}d_{Z}^{T})^{⋆}(β × γ))
= (^{X}p_{Y}^{T} )_{∗} ∘ (^{X}d_{Y}^{T} )^{⋆} ∘ (^{X×Y ×Y}p_{Z}^{T})_{∗} ∘ (^{X×Y ×Y}d_{Z}^{T})^{⋆}(α × β × γ)
= (^{X}p_{Y}^{T} )_{∗} ∘ (^{X×Y}p_{Z}^{T})_{∗} ∘ (^{X}d_{Y}^{Z×T} )^{⋆} ∘ (^{X×Y ×Y}d_{Z}^{T})^{⋆}(α × β × γ)
= (^{X}p_{Y ×Z}^{T})_{∗} ∘ (^{X}d_{Y ×Z}^{T})^{⋆}(α × β × γ).  □

Let f : X → Y be a morphism of schemes. The isomorphic image of X under the closed embedding (1_{X}, f) : X → X × Y is called the graph of f and is denoted by Γ_{f} . Thus, Γ_{f} is a closed subscheme of X × Y isomorphic to X under the projection X × Y → X. The class [Γ_{f}] belongs to CH(X × Y).

Proposition 62.4. Let X, Y, Z be schemes over F with Y smooth and complete. (1) For every morphism g : Y → Z and α ∈ A_{∗}(X × Y, K_{∗}),

[Γ_{g}] ∘ α = (1_{X} × g)_{∗}(α).
(2) For every morphism f : X → Y and β ∈ A_{∗}(Y × Z, K_{∗}),
β ∘ [Γ_{f}] = (f × 1_{Z})^{∗}(β).

Proof. (1): Consider the commutative diagram

X × Y ⟵^{^{X×Y}p_{Y}} X × Y × Y ⟵^{^{X}d_{Y}} X × Y
r↓  ^{t}↓
X × Y × Y × Z ⟵^{^{X}d_{Y}^{Z}} X × Y × Z ⟶^{^{X}p_{Y}^{Z}} X × Z

where r = 1_{X×Y} × (1_{Y} , g) and t = 1_{X} × (1_{Y} , g).

The composition ^{X×Y}p_{Y} ∘^{X}d_{Y} is the identity on X × Y and ^{X}p_{Y}^{Z} ∘ t = 1_{X} × g. It follows from Corollary 55.4 that (^{X}d_{Y}^{Z} )^{⋆} ∘ r_{∗} = t_{∗} ∘ (^{X}d_{Y} )^{⋆}. We have

[Γ_{g}] ∘ α = (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆}(α × [Γ_{g}])
= (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆} ∘ r_{∗}(α × [Y])
= (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆} ∘ r_{∗} ∘ (^{X×Y}p_{Y})^{∗}(α)
= (^{X}p_{Y}^{Z} )_{∗} ∘ t_{∗} ∘ (^{X}d_{Y} )^{⋆} ∘ (^{X×Y}p_{Y} )^{∗}(α)
= (1_{X} × g)_{∗}(α).

(2): Consider the commutative diagram

Y × Z ⟵^{p_{X}^{Y ×Z}} X × Y × Z ⟵^{v} X × Z
u↓  ^{v}↓
X × Y × Y × Z ⟵^{^{X}d_{Y}^{Z}} X × Y × Z ⟶^{^{X}p_{Y}^{Z}} X × Z

where u = (1_{X}, f) × 1_{Y ×Z} and v = (1_{X}, f) × 1_{Z}.

The composition ^{X}p_{Y}^{Z} ∘ v is the identity on X × Z and p_{X}^{Y ×Z} ∘ v = f × 1_{Z}. It follows from Corollary 55.4 that (^{X}d_{Y}^{Z} )^{⋆} ∘ u_{∗} = v_{∗} ∘ v^{∗}. We have

β ∘ [Γ_{f}] = (^{X}p_{Y}^{Z})_{∗} ∘ (^{X}d_{Y}^{Z})^{⋆}([Γ_{f}] × β)
= (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆} ∘ u_{∗}([X] × β)
= (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆} ∘ u_{∗} ∘ (p_{X}^{Y ×Z})^{∗}(β)
= (^{X}p_{Y}^{Z} )_{∗} ∘ v_{∗} ∘ v^{∗} ∘ (p_{X}^{Y ×Z})^{∗}(β)
= (f × 1_{Z})^{∗}(β).  □

Corollary 62.5. Let X and Y be schemes over F and α ∈ A_{∗}(X × Y, K_{∗}). Then (1) If Y is smooth and complete, then [Γ_{1_{Y}}] ∘ α = α.

(2) If X is smooth and complete, then α ∘ [Γ_{1_{X}}] = α.

Corollary 62.6. Let f : X → Y and g : Y → Z be two morphisms. If Y is smooth and complete, then [Γ_{g}] ∘ [Γ_{f}] = [Γ_{gf}].

Proof. By Proposition 62.4(1),

[Γ_{g}] ∘ [Γ_{f}] = (1_{X} × g)_{∗}([Γ_{f}])
= (1_{X} × g)_{∗}(1_{X}, f)_{∗}([X])
= (1_{X}, gf)_{∗}([X])
= [Γ_{gf}].  □

Let X, Y and Z be arbitrary schemes and α ∈ A_{∗}(X × Y, K_{∗}). If X is smooth and complete, we have a well-defined homomorphism

α_{∗} : A_{∗}(Z × X, K_{∗}) → A_{∗}(Z × Y, K_{∗}), β ↦ α ∘ β.

If α = [Γ_{f}] with f : X → Y a morphism, it follows from Proposition 62.4(1) that α_{∗} = (1_{Z} × f)_{∗}.

If X is smooth and Z = Spec(F), we get a homomorphism α_{∗} : A_{∗}(X, K_{∗}) → A_{∗}(Y, K_{∗}). We have simpler formula for α_{∗} in the following case:

Proposition 62.7. Let α = [T] with T ⊂ X × Y a closed subscheme with X smooth and complete. Then α_{∗} = q_{∗} ∘ p^{∗}, where p : T → X and q : T → Y are the projections.

Proof. Let r : X × Y → Y be the projection, i : T → X × Y the closed embedding, and f : T → X × T the graph of the projection p. Consider the commutative diagram

X × T ⟵^{f} T ⟶^{q} Y
1_{X}×i↓  ^{i}↓  ∥
X × X × Y ⟵^{d_{X}^{Y}} X × Y ⟶^{r} Y.

It follows from Corollary 55.4 that i_{∗} ∘ f^{∗} = (d_{X}^{Y})^{⋆} ∘ (1_{X} × i)_{∗}. Therefore, for every β ∈ A_{∗}(X, K_{∗}), we have

α_{∗}(β) = r_{∗} ∘ (d_{X}^{Y})^{⋆}(β × α)
= r_{∗} ∘ (d_{X}^{Y})^{⋆} ∘ (1_{X} × i)_{∗}(β × [T])
= r_{∗} ∘ i_{∗} ∘ f^{∗}(β × [T])
= q_{∗} ∘ f^{∗}(β × [T])
= q_{∗} ∘ p^{∗}(β).  □

If Y is smooth and complete, we have a well-defined homomorphism

α^{∗} : A_{∗}(Y × Z, K_{∗}) → A_{∗}(X × Z, K_{∗}), β ↦ β ∘ α.

If α = [Γ_{f}] for a flat morphism f : X → Y, it follows from Proposition 62.4(2) that α^{∗} = (f × 1_{Z})^{∗}.

Let X, Y and Z be arbitrary schemes, α ∈ A_{∗}(X × Y, K_{∗}) and g : Y → Z a proper morphism. We define the composition of g and α by

g ∘ α := (1_{X} × g)_{∗}(α) ∈ A_{∗}(X × Z, K_{∗}).

If g ∘ α = [Γ_{h}] for some morphism h : X → Z, we abuse notation and just write g ∘ α = h. If Y is smooth and complete, we have g ∘ α = [Γ_{g}] ∘ α by Proposition 62.4(1).

Similarly, if β ∈ A_{∗}(Y × Z, K_{∗}) and f : X → Y is a flat morphism, we define the composition of β and f by

β ∘ f := (f × 1_{Z})^{∗}(β) ∈ A_{∗}(X × Z, K_{∗}).

If Y is smooth and complete, we have β ∘ f = β ∘ [Γ_{f}] by Proposition 62.4(2).

The following statement is an analogue of Proposition 62.3 with fewer assumptions on the schemes.

Proposition 62.8. Let X, Y, Z and T be arbitrary schemes.

(1) Let α ∈ A_{∗}(X × Y, K_{∗}), γ ∈ A_{∗}(T × X, K_{∗}), and g : Y → Z a proper morphism. If X is smooth and complete, then

(g ∘ α) ∘ γ = g ∘ (α ∘ γ), i.e., (g ∘ α)_{∗} = g_{∗} ∘ α_{∗}.

(2) Let β ∈ A_{∗}(Y × Z, K_{∗}), δ ∈ A_{∗}(Z × T, K_{∗}), and f : X → Y a flat morphism. If Z is smooth and complete, then

δ ∘ (β ∘ f) = (δ ∘ β) ∘ f, i.e., (β ∘ f)^{∗} = f^{∗} ∘ β^{∗}.

Proof. (1): Consider the commutative diagram with fiber product squares

T × X × X × Y ⟵^{^{T}d_{X}^{Y}} T × X × Y ⟶^{^{T}p_{X}^{Y}} T × Y
1_{T ×X×X}×g↓  ^{1_{T ×X}×g}↓  ↓^{1_{T} ×g}
T × X × X × Z ⟵^{^{T}d_{X}^{Z}} T × X × Z ⟶^{^{T}p_{X}^{Z}} T × Z.

It follows from Proposition 50.4 and Corollary 55.4 that

g ∘ (α ∘ γ) = (1_{T} × g)_{∗}(α ∘ γ)
= (1_{T} × g)_{∗} ∘ (^{T}p_{X}^{Y})_{∗} ∘ (^{T}d_{X}^{Y})^{⋆}(γ × α)
= (^{T}p_{X}^{Z})_{∗} ∘ (^{T}d_{X}^{Z})^{⋆} ∘ (1_{T ×X×X} × g)_{∗}(γ × α)
= (1_{X} × g)_{∗}(α) ∘ γ
= (g ∘ α) ∘ γ.

(2): The proof is similar using Propositions 49.20, 50.5 and 55.5 instead. □

If γ ∈ A_{∗}(Y × X, K_{∗}) and g : Y → Z is a proper morphism, we write γ ∘ g^{t} for (g ∘ γ^{t})^{t} ∈ A_{∗}(Z × X, K_{∗}). Similarly, if δ ∈ A_{∗}(Z × Y, K_{∗}) and f : X → Y is a flat morphism, we define the composition f^{t} ∘ δ ∈ A_{∗}(Z × X, K_{∗}) to be (δ^{t} ∘ f)^{t}.

63. Categories of correspondences

Let Λ be a commutative ring. For a scheme Z, we write CH(Z; Λ) for the Λ-module CH(Z) ⊗ Λ.

Let X and Y be smooth complete schemes over F. Let X_{1}, X_{2}, …, X_{n} be the irreducible components of X of dimension d_{1}, d_{2}, …, d_{n}, respectively. For every i ∈ ℤ, we set

Corr_{i}(X, Y; Λ) := ∐_{k=1}^{n} CH_{i+d_{k}}(X_{k} × Y; Λ).

An element α ∈ Corr_{i}(X, Y; Λ) is called a correspondence between X and Y of degree i with coefficients in Λ. We write α : X ⇝ Y.

Let Z be another smooth complete scheme. By Proposition 62.3, the bilinear pairing (β, α) ↦ β ∘ α on Chow groups yields an associative pairing (composition)

(63.1)  Corr_{i}(Y, Z; Λ) × Corr_{j}(X, Y; Λ) → Corr_{i+j}(X, Z; Λ).

The following proposition gives an alternative formula for this composition involving only projection morphisms.

Proposition 63.2. β ∘ α = (^{X}p_{Y}^{Z} )_{∗}((^{X×Y}p_{Z})^{∗}(α) · (p_{X}^{Y ×Z})^{∗}(β)).

Proof. Let f : X × Y × Y × Z → X × Y × Z × X × Y × Z be defined by f(x, y, y′, z) = (x, y, z, x, y′, z). We have f ∘^{X}d_{Y}^{Z} = d_{X×Y ×Z}, therefore,

β ∘ α = (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆}(α × β)
= (^{X}p_{Y}^{Z} )_{∗} ∘ (^{X}d_{Y}^{Z} )^{⋆} ∘ f^{∗}(α × [Z] × [X] × β)
= (^{X}p_{Y}^{Z} )_{∗} ∘ (d_{X×Y ×Z})^{⋆}((^{X×Y}p_{Z})^{∗}(α) × (p_{X}^{Y ×Z})^{∗}(β))
= (^{X}p_{Y}^{Z} )_{∗}((^{X×Y}p_{Z})^{∗}(α) · (p_{X}^{Y ×Z})^{∗}(β)).  □

Define the category CR_{∗}(F, Λ) of correspondences with coefficients in Λ over F as follows: Objects of CR_{∗}(F, Λ) are smooth complete schemes over F. A morphism between X and Y is an element of the graded group

∐_{k∈ℤ} Corr_{k}(X, Y; Λ).

Composition of morphisms is given by (63.1). The identity morphism of X in CR_{∗}(F, Λ) is Γ_{1_{X}} ⊗ 1, where Γ_{1_{X}} is the class of the graph of the identity morphism 1_{X} (cf. Corollary 62.5). The direct sum in CR_{∗}(F, Λ) is given by the disjoint union of schemes. As the composition law in CR_{∗}(F, Λ) is bilinear and associative by Proposition 62.3, the category CR_{∗}(F, Λ) is additive. Abusing notation, we write Λ for the object Spec(F) in this category.

An object of CR_{∗}(F, Λ) is called a Chow motive or simply a motive. If X is a smooth complete scheme, we write M(X) for it as an object in CR_{∗}(F, Λ).

We define another category C(F, Λ) as follows. Objects of C(F, Λ) are pairs (X, i), where X is a smooth complete scheme over F and i ∈ ℤ. A morphism between (X, i) and (Y, j) is an element of Corr_{i−j}(X, Y; Λ). The composition of morphisms is given by (63.1). The morphisms between two objects form an abelian group and the composition is bilinear and associative by Proposition 62.3, therefore, C(F, Λ) is a preadditive category.

There is an additive functor C(F, Λ) → CR_{∗}(F, Λ) taking an object (X, i) to X, and that is the natural inclusion on morphisms.

Let A be a preadditive category. Define the additive completion of A to be the following category Ã: its objects are finite sequences of objects A_{1}, …, A_{n} of A written in the form ∐_{i=1}^{n} A_{i}. A morphism between ∐_{i=1}^{n} A_{i} and ∐_{j=1}^{m} B_{j} is given by an n × m-matrix of morphisms A_{i} → B_{j}. The composition of morphisms is the matrix multiplication. The category Ã has finite products and coproducts and is therefore an additive category. The category A is a full subcategory of Ã.

Let CR(F, Λ) denote the additive completion of C(F, Λ). We call it the category of graded correspondences with coefficients in Λ over F. An object of CR(F, Λ) is also called a Chow motive or simply a motive. We will write M(X)(i) for (X, i) and simply M(X) for (X, 0). The functor C(F, Λ) → CR_{∗}(F, Λ) extends naturally to an additive functor

(63.3)  CR(F, Λ) → CR_{∗}(F, Λ).

taking M(X)(i) to M(X). Write Λ(i) for the motive (Spec F, i) in CR(F, Λ). The motives Λ(i) in CR(F, Λ) and Λ in CR_{∗}(F, Λ) are called the Tate motives.

Functor (63.3) is faithful but not full. Nevertheless, it has the following nice property.

Proposition 63.4. Let f be a morphism in CR(F, Λ). If the image of f in the category CR_{∗}(F, Λ) is an isomorphism, then f itself is an isomorphism.

Proof. Let f be a morphism between the objects

∐_{i=1}^{n} X_{i}(a_{i}) and ∐_{j=1}^{m} Y_{j}(b_{j}).

Thus f is given by an n × m matrix A = (f_{ij}) with f_{ij} ∈ Corr_{a_{j}−b_{i}}(X_{j}, Y_{i}; Λ). Let B = (g_{kl}) be the matrix of the inverse of f in CR_{∗}(F, Λ), so that g_{kl} ∈ Corr_{∗}(Y_{l}, X_{k}; Λ). Let ḡ_{kl} be the homogeneous component of g_{kl} of degree b_{l} − a_{k} and \overline{B} = (ḡ_{kl}). As AB = A\overline{B} and \overline{B}A = BA are the identity matrices, we have \overline{B} = B = A^{−1}. Therefore, B is the matrix of the inverse of f in CR(F, Λ). □

A ring homomorphism Λ → Λ′ gives rise to natural functors CR_{∗}(F, Λ) → CR_{∗}(F, Λ′) and CR(F, Λ) → CR(F, Λ′) that are identical on objects. We simply write CR_{∗}(F) for CR_{∗}(F, ℤ) and CR(F) for CR(F, ℤ).

It follows from Corollary 62.6 that there is a functor

Sm(F) → CR(F, Λ)

taking a smooth complete scheme X to M(X) and a morphism f : X → Y to [Γ_{f}] ⊗ 1 in Corr_{0}(X, Y; Λ) = Mor_{CR(F,Λ)}(M(X), M(Y)), where Γ_{f} is the graph of f.

Let X and Y be smooth complete schemes and i, j ∈ ℤ. We have

Hom_{CR(F)}(M(X)(i), M(Y)(j)) = Corr_{i−j}(X, Y; Λ).

In particular,

(63.5)  Hom_{CR(F,Λ)}(Λ(i), M(X)) = CH_{i}(X; Λ),
(63.6)  Hom_{CR(F,Λ)}(M(X), Λ(i)) = CH^{i}(X; Λ).

We define Chow groups with coefficients in Λ for an arbitrary motive M as follows:

CH_{i}(M; Λ) := Hom_{CR(F,Λ)}(Λ(i), M), CH^{i}(M; Λ) := Hom_{CR(F,Λ)}(M, Λ(i)).

The category CR(F, Λ) has the structure of a tensor category given by

M(X)(i) ⊗ M(Y)(j) := M(X × Y)(i + j).

In particular,

M(X)(i) ⊗ Λ(j) = M(X)(i + j).

Let Y be a smooth variety of dimension d. By the definition of a morphism in CR(F), the equality

(63.7)  Hom_{CR(F,Λ)}(M(Y)(i), N) = CH_{d+i}(M(Y) ⊗ N; Λ)

holds for every N of the form M(X)(j), where X is a smooth complete scheme; and, therefore, by additivity it holds for all motives N. Similarly,

(63.8)  Hom_{CR(F,Λ)}(N, M(Y)(i)) = CH^{d+i}(N ⊗ M(Y); Λ).

The following statement is a variant of the Yoneda Lemma.

Lemma 63.9. Let α : N → P be a morphism in CR(F, Λ). Then the following conditions are equivalent:

(1) α is an isomorphism.

(2) For every smooth complete scheme Y, the homomorphism

(1_{Y} ⊗ α)_{∗} : CH_{∗}(M(Y) ⊗ N; Λ) → CH_{∗}(M(Y) ⊗ P; Λ)

is an isomorphism.

(3) For every smooth complete scheme X, the homomorphism

(1_{Y} ⊗ α)^{∗} : CH^{∗}(M(Y) ⊗ P; Λ) → CH^{∗}(M(Y) ⊗ N; Λ)

is an isomorphism.

Proof. Clearly (1) ⇒ (2) and (1) ⇒ (3). We prove that (2) implies (1) (the proof of the implication (3) ⇒ (1) is similar and left to the reader). It follows from (63.7) that the natural homomorphism

Hom_{CR(F,Λ)}(M, N) → Hom_{CR(F,Λ)}(M, P)

is an isomorphism if M = M(Y)(i) for any smooth complete variety Y. By additivity, it is isomorphism for all motives M. The statement now follows from the (usual) Yoneda Lemma. □

The following statement is the motivic version of the Projective Bundle Theorem 53.10.

Theorem 63.10. Let E → X be a vector bundle of rank r over a smooth complete scheme X. Then the motives M(ℙ(E)) and ∐_{i=0}^{r−1} M(X)(i) are naturally isomorphic in CR(F, Λ).

Proof. Let Y be a smooth complete scheme over F. Applying the Projective Bundle Theorem 53.10 to the vector bundle E × Y → X × Y, we see that the Chow groups of ∐_{i=0}^{r−1} M(X × Y)(i) and M(ℙ(E) × Y ) are isomorphic. Moreover, in view of Remark 53.11, this isomorphism is natural in Y with respect to morphisms in the category CR(F, Λ). In other words, the functors on CR(F, Λ) represented by the objects ∐_{i=0}^{r−1} M(X)(i) and M(ℙ(E)) are isomorphic. By the Yoneda Lemma, the objects are isomorphic in CR(F, Λ). □

Corollary 63.11. The motive M(ℙ(E)) is isomorphic to the direct sum M(X)^{r} of r copies of M(X) in the category CR_{∗}(F, Λ).

64. Category of Chow motives

Let A be an additive category. An idempotent e : A → A in A is called split if there is an isomorphism f : A →^{∼} B ⊕ C such that e coincides with the composition

A ⟶^{f} B ⊕ C ⟶^{p} B ⟶^{i} B ⊕ C ⟶^{f^{−1}} A, where p and i are the canonical morphisms.

The idempotent completion of an additive category A is the category \overline{A} defined as follows. Objects of \overline{A} are the pairs (A, e), where A is an object of A and e : A → A is an idempotent. The group of morphisms between (A, e) and (B, f) is f ∘ Hom_{A}(A, B) ∘ e. Every idempotent in \overline{A} is split.

The assignment A ↦ (A, 1_{A}) defines a full and faithful functor from A to \overline{A}. We identify A with a full subcategory of \overline{A}.

Let Λ be a commutative ring. The idempotent completion of the category CR(F, Λ) is called the category of graded Chow motives with coefficients in Λ and is denoted by CM(F, Λ). By definition, every object of CM(F, Λ) is a direct summand of a finite direct sum of motives of the form M(X)(i), where X is a smooth complete scheme over F. We write CM(F) for CM(F, ℤ).

Similarly, the idempotent completion CM_{∗}(F, Λ) of CR_{∗}(F, Λ) is called the category of Chow motives with coefficients in Λ. Note that Proposition 63.4 holds for the natural functor CM(F, Λ) → CM_{∗}(F, Λ).

We have functors

Sm(F) → CR(F, Λ) → CM(F, Λ).

The second functor is full and faithful, so we can view CR(F, Λ) as a full subcategory of CM(F, Λ) which we do. Note that CM(F, Λ) inherits the structure of a tensor category.

An object of CM(F, Λ) is also called a motive. We will keep the same notation M(X)(i), Λ(i), etc. for the corresponding motives in CM(F, Λ). The motives Λ(i) and Λ are called the Tate motives.

We use formulas (63.5) and (63.6) in order to define Chow groups with coefficients in Λ for an arbitrary motive M:

CH_{i}(M; Λ) := Hom_{CM(F,Λ)}(Λ(i), M), CH^{i}(M; Λ) := Hom_{CM(F,Λ)}(M, Λ(i)).

The functor from CM(F, Λ) to the category of Λ-modules, taking a motive M to CH_{i}(M; Λ) (respectively the cofunctor M ↦ CH^{i}(M; Λ)) is then represented (respectively corepresented) by Λ(i).

Let Y be a smooth variety of dimension d. It follows from (63.7) and (63.8) that

(64.1)  Hom_{CM(F,Λ)}(M(Y)(i), N) = CH_{d+i}(M(Y) ⊗ N; Λ),
Hom_{CM(F,Λ)}(N, M(Y)(i)) = CH^{d+i}(N ⊗ M(Y); Λ).

for all motives N in CM(F, Λ).

Let M and N be objects in CM(F). The tensor product of two morphisms M → Λ(i) and N → Λ(j) defines a pairing

(64.2)  CH^{∗}(M; Λ) ⊗ CH^{∗}(N; Λ) → CH^{∗}(M ⊗ N; Λ).

Note that this is an isomorphism whenever M (or N) is a Tate motive.

We say that an object M of CR(F, Λ) is split if M is isomorphic to a (finite) coproduct of Tate motives. The additivity property of the pairing yields

Proposition 64.3. Let M (or N) be a split motive. Then the homomorphism (64.2) is an isomorphism.

65. Duality

There is an additive duality functor ∗ : CM(F, Λ)^{op} → CM(F, Λ) uniquely determined by the rule M(X)(i)^{∗} = M(X)(−d − i) where X is a smooth complete variety of dimension d and α^{∗} = α^{t} for any correspondence α. In particular, Λ(i)^{∗} = Λ(−i). The composition ∗ ∘ ∗ is the identity functor.

It follows from the definition of the duality functor that

Hom_{CM(F,Λ)}(M^{∗}, N^{∗}) = Hom_{CM(F,Λ)}(N, M)

for every two motives M and N. In particular, setting N = Λ(i), we get

CH^{i}(M^{∗}; Λ) = CH_{−i}(M; Λ).

The equality (64.1) reads as follows:

(65.1)  Hom_{CM(F,Λ)}(M(Y)(i), N) = CH_{0}(M(Y)(i)^{∗} ⊗ N; Λ)

for every smooth complete scheme Y. Set

\underline{Hom}(M, N) := M^{∗} ⊗ N

for every two motives M and N. By additivity, the equality (65.1) yields

Hom_{CM(F,Λ)}(M, N) = CH_{0}(\underline{Hom}(M, N); Λ).

Since the duality functor commutes with the tensor product, the definition of \underline{Hom} satisfies the associativity law

\underline{Hom}(M ⊗ N, P) = \underline{Hom}(M, \underline{Hom}(N, P))

for all motives M, N and P. Applying CH_{0} we get

Hom_{CM(F,Λ)}(M ⊗ N, P) = Hom_{CM(F,Λ)}(M, \underline{Hom}(N, P)).

66. Motives of cellular schemes

Motives of cellular schemes were considered in [70]. Recall that a morphism p : U → Y over F is an affine bundle of rank d if f is flat and the fiber of p over any point y ∈ Y is isomorphic to the affine space 𝔸_{F (y)}^{d}.

A scheme X over F is called (relatively) cellular if there is given a filtration by closed subschemes

(66.1)  ∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} = X

together with affine bundles p_{i} : U_{i} := X_{i} \ X_{i−1} → Y_{i} of rank d_{i} for all i ∈ [0, n], where Y_{i} is a smooth complete scheme for all i ∈ [1, n] (we don’t assume that Y_{0} is either smooth or complete). The scheme U_{i} is called a cell of X and Y_{i} is the base of the cell U_{i}.

The graph Γ_{p_{i}} of the morphism p_{i} is a subscheme of U_{i} ×Y_{i}. Let α_{i} in CH(X_{i} × Y_{i}) be the class of the closure of Γ_{p_{i}} in X_{i} × Y_{i}. We have α_{0} = Γ_{p_{0}}.

We view α_{i} as a correspondence X_{i} ⇝ Y_{i} of degree 0. Let f_{i} : X_{i} → X be the closed embedding. The correspondence f_{i} ∘ α_{i}^{t} ∈ CH(Y_{i} × X) between Y_{i} and X is well-defined and of degree d_{i} for all i ≥ 1.

Let Z be a scheme over F. If h is a morphism of schemes, we still write h for the morphism 1_{Z} × h. We define homomorphisms

a_{i} : CH_{∗−d_{i}}(Z × Y_{i}) → CH_{∗}(Z × X)

for all i ∈ [0, n] as follows: We set a_{0} = (f_{0})_{∗} ∘ (p_{0}^{∗})^{−1} and a_{i} = (f_{i} ∘ α_{i}^{t})_{∗} if i ≥ 1. If Y_{0} is smooth and complete, then a_{0} = (f_{0} ∘ α_{0}^{t})_{∗}.

Theorem 66.2. Let X be a cellular scheme with filtration (66.1). Then for every scheme Z over F, the homomorphism

(66.3)  ∑_{i=0}^{n} a_{i} : ∐_{i=0}^{n} CH_{∗−d_{i}}(Z × Y_{i}) → CH_{∗}(Z × X)

is an isomorphism.

Proof. Let g_{i} : U_{i} → X_{i} denote the open embedding. By the definition of α_{i}, we have α_{i} ∘ g_{i} = p_{i}. It follows from Proposition 62.8(2) that for every scheme Z, the composition

A(Y_{i} × Z, K_{∗}) ⟶^{α_{i}^{∗}} A(X_{i} × Z, K_{∗}) ⟶^{g_{i}^{∗}} A(U_{i} × Z, K_{∗})

coincides with the pull-back homomorphism p_{i}^{∗} for i ≥ 1. By Theorem 52.13, the map p_{i}^{∗} is an isomorphism. Hence g_{i}^{∗} is a split surjection. Therefore, in the localization exact sequence (cf. §52.D),

A_{k+1}(X_{i} × Z, K_{−k}) ⟶^{g_{i}^{∗}} A_{k+1}(U_{i} × Z, K_{−k}) ⟶^{δ}

CH_{k}(X_{i−1} × Z) → CH_{k}(X_{i} × Z) ⟶^{g_{i}^{∗}} CH_{k}(U_{i} × Z) → 0,

the connecting homomorphism δ is trivial. Consequently, we have a short exact sequence

0 → CH(X_{i−1} × Z) → CH(X_{i} × Z) ⟶^{s_{i}} CH(Y_{i} × Z) → 0

where s_{i} = p_{i}^{∗}^{−1}∘g_{i}^{∗} and s_{i} is split by α_{i}^{∗} : CH(Y_{i}×Z) → CH(X_{i}×Z). In particular, CH(X_{i} × Z) is isomorphic to CH(X_{i−1} × Z) ⊕ CH(Y_{i} × Z). Iterating, we see that CH(X × Z) is isomorphic to the coproduct of the CH(Y_{i} × Z) over all i ∈ [0, n]. The inclusion of CH(Y_{i} × Z) into CH(X × Z) coincides with the composition

CH(Y_{i} × Z) ⟶^{α_{i}^{∗}} CH(X_{i} × Z) ⟶^{(f_{i})_{∗}} CH(X × Z),

where α_{0}^{∗} is understood to be p_{0}^{∗}. If i ≥ 1, we have a_{i} = (f_{i})_{∗} ∘ (α_{i}^{t})_{∗} by Proposition 62.8(1). Under the identification of CH(Y_{i} × Z) with CH(Z × Y_{i}), we have (α_{i}^{t})_{∗} = α_{i}^{∗}, hence a_{i} = (f_{i})_{∗} ∘ α_{i}^{∗}. It follows that the homomorphism (66.3) is an isomorphism. □

Lemma 63.9 yields

Corollary 66.4. Let X be a smooth complete cellular scheme with filtration (66.1) and with all Y_{i} smooth complete. Then the morphism

∐_{i=0}^{n} M(Y_{i})(d_{i}) → M(X),

defined by the sequence of correspondences f_{i} ∘α_{i}^{t} is an isomorphism in the category of correspondences CR(F).

Example 66.5. Let X = ℙ^{n}. Consider the filtration given by X_{i} = ℙ^{i}, i ∈ [0, n]. We have U_{i} = 𝔸^{i} and Y_{i} = Spec(F). By Corollary 66.4,

M(ℙ^{n}) = ℤ ⊕ ℤ(1) ⊕ ⋯ ⊕ ℤ(n).

Example 66.6. Let φ be a nondegenerate quadratic form on V and X the associated quadric of say dimension d. Consider the following filtration on X × X: X_{0} is the image of the diagonal embedding of X into X × X, X_{1} consists of all pairs of orthogonal isotropic lines (L_{1}, L_{2}), and X_{2} = X × X. We also set Y_{0} = X (with the identity projection of X_{0} on Y_{0}), Y_{2} = X, and Y_{1} is the flag variety Fl of pairs (L, P), where L and P are a totally isotropic line and plane, respectively, satisfying L ⊂ P.

We claim that the morphism p_{1} : U_{1} → Y_{1} taking a pair (L_{1}, L_{2}) to (L_{1}, L_{1}+L_{2}) is an affine bundle. To do this we use the criterion of Lemma 52.12. Let R be a local commutative F -algebra. An R-point of Y_{1} is a pair (L, P), where L and P are totally isotropic direct summands of the R-module V_{R} := V ⊗_{F} R of rank 1 and 2, respectively, with L ⊂ P. Let {e, f} be an R-basis of P such that L = Re. Then the morphism 𝔸_{R}^{1} → Spec(R) ×_{Y_{1}} U_{1} taking an a to the point (L, R(ae + f)) of the fiber product is an isomorphism. It follows from Lemma 52.12 that p_{1} is an affine bundle.

We claim that the first projection p_{2} : U_{2} → Y_{2} is an affine bundle of rank d. We again apply the criterion of Lemma 52.12. Let R be a local commutative F -algebra. An R-point of Y_{2} is a totally isotropic direct summand L ⊂ V_{R} of rank 1. Choose a basis of V_{R} so that φ_{R} is given by a polynomial t_{0}t_{1} + ψ(T′), where ψ is a quadratic form in the variables T′ = (t_{2}, …, t_{d+1}) over R, and the orthogonal complement L^{⊥} is given by t_{0} = 0. Then the fiber product Spec(R) ×_{Y_{2}} U_{2} is given by the equation \frac{t_{1}}{t_{0}} + ψ(\frac{T′}{t_{0}} ) = 0 and therefore is isomorphic to 𝔸_{R}^{d}. It follows by Lemma 52.12 that p_{2} is an affine bundle.

By Corollary 66.4, we conclude that

M(X × X) ≃ M(X) ⊕ M(Fl)(1) ⊕ M(X)(d).

Example 66.7. Assume that the quadric X in Example 66.6 is isotropic. The cellular structure on X × X is a structure “over X” in the sense that X × X itself as well as the bases Y_{i} of the cells have morphisms to X with affine bundles of the cellular structure morphisms over X. Making the base change of the cellular structure with respect to an F -point Spec(F) → X of the isotropic quadric X corresponding to an isotropic line L, we get a cellular structure on X given by the filtration X′_{0} ⊂ X′_{1} ⊂ X′_{2} = X, where X′_{0} = {L} and X′_{1} consists of all isotropic lines orthogonal to L. We have Y′_{0} = Spec(F), Y′_{1} is the quadric given by the quadratic form on L^{⊥}/L induced by φ, and Y′_{2} = Spec(F). The quadric Y′_{1} is isomorphic to a projective quadric Y of dimension d − 2, given by a quadratic form Witt equivalent to φ. By Corollary 66.4,

M(X) ≃ ℤ ⊕ M(Y)(1) ⊕ ℤ(d).

67. Nilpotence Theorem

Let Λ be a commutative ring and Y a smooth complete scheme over F. For every scheme X and elements α ∈ CH(Y × Y; Λ) and β ∈ CH(X × Y; Λ), the compositions α^{k} = α ∘ ⋯ ∘ α in CH(Y × Y; Λ) and α^{k} ∘ β in CH(X × Y; Λ) are defined.

Theorem 67.1 (Nilpotence Theorem). Let Y be a smooth complete scheme and X a scheme of dimension d over F. Let α ∈ CH(Y ×Y; Λ) be an element satisfying α ∘ CH(Y_{F (x)}; Λ) = 0 for every x ∈ X. Then

α^{d+1} ∘ CH(X × Y; Λ) = 0.

Proof. Consider the filtration

0 = C_{−1} ⊂ C_{0} ⊂ ⋯ ⊂ C_{d} = CH(X × Y; Λ),

where C_{i} is the Λ-submodule of CH(X × Y; Λ) generated by the images of the push-forward homomorphisms

CH(W × Y; Λ) → CH(X × Y; Λ),

for all closed subvarieties W ⊂ X of dimension at most k. It suffices to prove that α ∘ C_{k} ⊂ C_{k−1} for all k ∈ [0, d].

Let W be a closed subvariety of X of dimension k. Denote by i : W → X the closed embedding and by w the generic point of W. Pick any element β ∈ CH(W ×Y; Λ). We shall prove that α∘(i_{∗}(β)) ∈ C_{k−1}. Let β_{w} be the pull-back of β under the canonical morphism Y_{F (w)} → W ×Y. By assumption, α∘β_{w} = 0. By the Continuity Property 52.9, there is a nonempty open subscheme U (a neighborhood of w) in W satisfying α ∘ (β|_{U×Y}) = 0. It follows by Proposition 62.8(2) that

(α ∘ β)|_{U×Y} = α ∘ (β|_{U×Y}) = 0.

The complement V of U in W is a closed subscheme of W of dimension less than k. It follows from the exactness of the localization sequence (cf. Proposition 57.9)

CH(V × Y; Λ) → CH(W × Y; Λ) → CH(U × Y; Λ) → 0

that α ∘ β belongs to the image of the first map in this sequence. Therefore, the push-forward of the element α ∘ β in CH(X × Y; Λ) lies in the image of the pushforward homomorphism

CH(V × Y; Λ) → CH(X × Y; Λ).

Consequently, α ∘ (i_{∗}β) = α ∘ (β ∘ i^{t}) = (α ∘ β) ∘ i^{t} = (i × 1_{Y})_{∗}(α ∘ β) ∈ C_{k−1}. □

The Nilpotence Theorem 67.1 was originally proven by Rost using the cycle modules technique.

Part 3

Quadratic forms and algebraic cycles

CHAPTER XIII Cycles on Powers of Quadrics

Throughout this chapter, F is a field (of an arbitrary characteristic) and, in all sections but §70 and §71, X is a smooth projective quadric over F of even dimension D = 2d ≥ 0 or of odd dimension D = 2d+1 ≥ 1 given by a nondegenerate quadratic form φ on a vector space V over F (of dimension D + 2). For any integer r ≥ 1, we write X^{r} for the direct product X × ⋯ × X (over F) of r copies of X.

In this chapter, we study algebraic cycles on X^{r} for all r. We also obtain many results for the special case r = 2. In order to make their statements and proofs more accessible, we introduce certain diagrams of cycles on X^{2} (cf. §73).

68. Split quadrics

In this section the quadric X will be split, i.e., the Witt index 𝔦_{0}(X) has the maximal value d + 1.

Let V be the underlying vector space of φ. Fix a maximal totally isotropic subspace W ⊂ V. We write ℙ(V) for the projective space of V; this is the projective space in which the quadric X lies as a hypersurface. Note that the subspace ℙ(W) of ℙ(V) is contained in X.

Proposition 68.1. Let h ∈ CH^{1}(X) be the pull-back of the hyperplane class in CH^{1}(ℙ(V)). For any integer i ∈ [0, d], let l_{i} ∈ CH_{i}(X) be the class of an idimensional subspace of ℙ(W). Then the total Chow group CH(X) is free with basis {h^{i}, l_{i}| i ∈ [0, d]}. Moreover, the following multiplication rule holds in the ring CH(X):

h · l_{i} = l_{i−1} for any i ∈ [1, d].

Proof. Let W^{⊥} be the orthogonal complement of W in V (clearly, W^{⊥} = W if D is even; otherwise, W^{⊥} contains W as a hyperplane). The quotient map V → V/W^{⊥} induces a morphism X \ ℙ(W) → ℙ(V/W^{⊥}), which is an affine bundle of rank D − d. Therefore, by Theorem 66.2,

CH_{i}(X) ≃ CH_{i}(ℙ(W)) ⊕ CH_{i−D+d}(ℙ(V/W^{⊥}))

for any i, where the injection CH_{∗}(ℙ(W)) ↪ CH_{∗}(X) is the push-forward with respect to the embedding ℙ(W) ↪ X.

To better understand the second summand in the decomposition of CH(X), we note that the reduced intersection of ℙ(W^{⊥}) with X in ℙ(V) is ℙ(W), and that the affine bundle X\ℙ(W) → ℙ(V/W^{⊥}) above is the composite of the closed embedding X \ ℙ(W) ↪ ℙ(V) \ ℙ(W^{⊥}) with the evident vector bundle ℙ(V) \ ℙ(W^{⊥}) → ℙ(V/W^{⊥}). It follows that for any i ≤ d, the image of CH^{i}(ℙ(V/W^{⊥})) in CH^{i}(X) coincides with the image of the pull-back CH^{i}(ℙ(V)) → CH^{i}(X) (which is generated by h^{i}).

To check the multiplication formula, we consider the closed embeddings f : ℙ(W) ↪ X and g : X ↪ ℙ(V). Write L_{i} for the class in CH(ℙ(W)) of an idimensional linear subspace of ℙ(W), and H for the hyperplane class in CH(ℙ(V)). Since h = g^{∗}(H) and l_{i} = f_{∗}(L_{i}), we have by the projection formula (Proposition 56.9) and functoriality of the pull-back (Proposition 55.18),

h · l_{i} = g^{∗}(H) · f_{∗}(L_{i}) = f_{∗}((g ∘ f)^{∗}(H) · L_{i}).

By Corollary 57.20 (together with Propositions 104.16 and 55.19), we see that (g ∘ f)^{∗}(H) is the hyperplane class in CH(ℙ(W)); hence (g ∘ f)^{∗}(H) · L_{i} = L_{i−1} by Example 57.23. □

Proposition 68.2. For each i ∈ [0, D/2), the i-dimensional subspaces of ℙ(V) lying inside of X have the same class in CH_{i}(X). If D is even, there are precisely two different classes of d-dimensional subspaces and the sum of these two classes is equal to h^{d}.

Proof. By Proposition 68.1, the push-forward homomorphism CH_{i}(X) → CH_{i}(ℙ(V)) is injective (even bijective) if i ∈ [0, D/2). Since the i-dimensional linear subspaces of ℙ(V) have the same class in CH(ℙ(V)), the first statement of Proposition 68.2 follows.

Assume that D is even. Then {h^{d}, l_{d}} is a basis for the group CH_{d}(X), where l_{d} is the class of the special linear subspace ℙ(W) ⊂ X. Let l′_{d} ∈ CH_{d}(X) be the class of an arbitrary d-dimensional linear subspace of X. Since l_{d} and l′_{d} have the same image under the push-forward homomorphism CH_{d}(X) → CH_{d}(ℙ(V)) whose kernel is generated by h^{d} − 2l_{d}, one has l′_{d} = l_{d} + n(h^{d} − 2l_{d}) for some n ∈ ℤ. Since there exists a linear automorphism of X moving l_{d} to l′_{d}, and h^{d} is, of course, invariant with respect to any linear automorphism, h^{d} and l′_{d} also form a basis for CH_{d}(X); consequently, the determinant of the matrix

(\begin{matrix}1 & n \\ 0 & 1 − 2n \end{matrix})

is ±1, i.e., n is 0 or 1 and l′_{d} is l_{d} or h^{d} − l_{d}. So there are at most two different rational equivalence classes of d-dimensional linear subspaces of X and the sum of two different classes (if they exist) is equal to h^{d}.

Now let U be a d-codimensional subspace of V containing W (as a hyperplane). The orthogonal complement U^{⊥} has codimension 1 in W^{⊥} = W; therefore codim_{U} U^{⊥} = 2. The induced 2-dimensional quadratic form on U/U^{⊥} is a hyperbolic plane. The corresponding quadric consists of two points W/U^{⊥} and W′/U^{⊥} for a uniquely determined maximal totally isotropic subspace W′ ⊂ V. Moreover, the intersection X ∩ ℙ(U) is reduced and its irreducible components are ℙ(W) and ℙ(W′). Therefore, h^{d} = [X ∩ ℙ(U)] = [ℙ(W)] + [ℙ(W′)] and it follows that [ℙ(W)] ≠ [ℙ(W′)]. □

Exercise 68.3. Determine a complete multiplication table for CH(X) by showing that

(1) h^{d+1} = 2l_{D−d−1};

(2) if D is not divisible by 4, then l_{d}^{2} = 0;

(3) if D is divisible by 4, then l_{d}^{2} = l_{0}.

Exercise 68.4. Assume that D is even and let l_{d}, l′_{d} ∈ CH(X) be two different classes of d-dimensional subspaces. Let f be the automorphism of CH(X) induced by a reflection. Show that f(l_{d}) = l′_{d}.

If D is even, an orientation of the quadric is the choice of one of two classes of d-dimensional linear subspaces in CH(X). We denote this class by l_{d}. An evendimensional quadric with an orientation is called oriented.

Proposition 68.5. For any r ≥ 1, the Chow group CH(X^{r}) is free with basis given by the external products of the basis elements {h^{i}, l_{i}}, i ∈ [0, d], of CH(X).

Proof. The cellular structure on X, constructed in the proof of Proposition 68.1, together with the calculation of the Chow motive of a projective space (cf. Example 66.5) show by Corollary 66.4 that the motive of X is split. Therefore, the homomorphism CH(X)^{⊗r} → CH(X^{r}), given by the external product of cycles is an isomorphism by Proposition 64.3. □

69. Isomorphisms of quadrics

Let φ and ψ be two quadratic forms. A similitude between φ and ψ (with multiplier a ∈ F^{×}) is an isomorphism f : V_{φ} → V_{ψ} such that φ(v) = aψ(f(v)) for all v ∈ V_{φ}. A similitude between φ and ψ induces an isomorphism of projective spaces ℙ(V_{φ}) →^{∼} ℙ(V_{ψ}) and projective quadrics X_{φ} →^{∼} X_{ψ}.

Let i : X_{φ} → ℙ(V_{φ}) be the embedding. We consider the locally free sheaves

O_{X_{φ}}(s) := i^{∗}(O_{ℙ(V_{φ})}(s))

over X_{φ} for every s ∈ ℤ.

Lemma 69.1. Let φ be a nonzero quadratic form of dimension at least 2. Then H^{0}(X_{φ}, O_{X_{φ}}(−1)) = 0 and H^{0}(X_{φ}, O_{X_{φ}}(1)) is canonically isomorphic to V_{φ}^{∗}.

Proof. We have H^{0}(ℙ(V_{φ}), O_{ℙ(V_{φ})}(−1)) = 0, H^{0}(ℙ(V_{φ}), O_{ℙ(V_{φ})}(1)) ≃ V_{φ}^{∗} and H^{1}(ℙ(V_{φ}), O_{ℙ(V_{φ})}(s)) = 0 for any s (see [50, Ch. III, Th. 5.1]). The statements follow from exactness of the cohomology sequence for the short exact sequence

0 → O_{ℙ(V_{φ})}(s − 2) ⟶^{φ} O_{ℙ(V_{φ})}(s) → i_{∗}O_{X_{φ}}(s) → 0.  □

Lemma 69.2. Let α : X_{φ} →^{∼} X_{ψ} be an isomorphism of smooth projective quadrics. Then α^{∗}(O_{X_{ψ}}(1)) ≃ O_{X_{φ}}(1).

Proof. In the case dim φ = 2 the sheaves O_{X_{φ}}(1) and O_{X_{ψ}}(1) are free and the statement is obvious.

We may assume that dim φ > 2. As the Picard group of smooth projective varieties injects under field extensions, we also may assume that both forms are split. We identify the groups Pic(X_{φ}) and CH^{1}(X_{φ}). The class of the sheaf O_{X_{φ}}(1) corresponds to the class h ∈ CH^{1}(X_{φ}) of a hyperplane section. It is sufficient to show that α^{∗}(h) = ±h, since the class −h cannot occur as the sheaf O_{X_{φ}}(−1) has no nontrivial global sections by Lemma 69.1.

If dim φ > 4 then, by Proposition 68.1, the group CH^{1}(X_{φ}) is infinite cyclic and generated by h. Thus α^{∗}(h) = ±h.

If dim φ = 3, then h is twice the generator l_{0} of the infinite cyclic group CH^{1}(X_{φ}) and the result follows in a similar fashion.

Finally, if dim φ = 4, then the group CH^{1}(X_{φ}) is a free abelian group with two generators l_{1} and l′_{1} satisfying l_{1} + l′_{1} = h (cf. the proof of Proposition 68.2). Using the fact that the pull-back map α^{∗} : CH(X_{φ}) → CH(X_{φ}) is a ring homomorphism, one concludes that α^{∗}(l_{1} + l′_{1}) = ±(l_{1} + l′_{1}). □

Theorem 69.3. Every isomorphism between smooth projective quadrics X_{φ} and X_{ψ} is induced by a similitude between φ and ψ.

Proof. Let α : X_{φ} →^{∼} X_{ψ} be an isomorphism. By Lemma 69.2, α^{∗}(O_{X_{ψ}}(1)) ≃ O_{X_{φ}}(1). Lemma 69.1 therefore gives an isomorphism of vector spaces

(69.4)  V_{ψ}^{∗} = H^{0}(X_{ψ}, O_{X_{ψ}}(1)) →^{∼} H^{0}(X_{φ}, O_{X_{φ}}(1)) = V_{φ}^{∗}.

Thus α is given by the induced graded ring isomorphism S^{•}(V_{ψ}^{∗}) → S^{•}(V_{φ}^{∗}) which must take the ideal (ψ) to (φ); i.e., it takes ψ to a multiple of φ. In other words, the linear isomorphism f : V_{φ} → V_{ψ} dual to (69.4) is a similitude between φ and ψ inducing α. □

Corollary 69.5. Let φ and ψ be nondegenerate quadratic forms. The quadrics X_{φ} and X_{ψ} are isomorphic if and only if φ and ψ are similar.

For a quadratic form φ, all similitudes V_{φ} → V_{φ} form the group of similitudes GO(φ). For every a ∈ F^{×}, the endomorphism of V_{φ} given by the product with a is a similitude. Therefore F^{×} can be identified with a subgroup of GO(φ). The factor group PGO(φ) := GO(φ)/F^{×} is called the group of projective similitudes. Every projective similitude induces an automorphism of the quadric X_{φ}, so we have a group homomorphism PGO(φ) → Aut(X_{φ}).

Corollary 69.6. Let φ be a nondegenerate quadratic form. Then the map

PGO(φ) → Aut(X_{φ})

is an isomorphism.

70. Isotropic quadrics

The motive of a smooth isotropic quadric is computed in terms of a smooth quadric of smaller dimension in Example 66.7 as follows:

Proposition 70.1. Let X be a smooth isotropic projective quadric given by a quadratic form φ with dim φ ≥ 2 and let Y be a (smooth) projective quadric given by a quadratic form of dimension dim φ − 2 that is Witt equivalent to φ (we assume that Y = ∅ if dim X ≤ 1). Then

M(X) ≃ ℤ ⊕ M(Y)(1) ⊕ ℤ(D)

with D = dim X. In particular,

CH_{∗}(X) ≃ CH_{∗}(ℤ) ⊕ CH_{∗−1}(Y) ⊕ CH_{∗−D}(ℤ).

The motivic decomposition of Proposition 70.1 was originally observed by Rost. In most of this book we are interested only in smooth quadrics. Nevertheless,

we make some observations concerning the general (not necessarily smooth) case. Let φ be a quadratic form on a vector space V over F and let X be the associated projective quadric. Suppose a rational point x on X is singular. Then φ is nonzero and x is determined by a nonzero vector v ∈ V in the radical of φ (cf. proof of Proposition 22.1). Let Y be the quadric given by the quadratic form on V/F v induced by φ, so dim Y = dim X − 1 and X \ {x} is an affine bundle over Y of rank 1. Since the first homomorphism in the exact localization sequence

CH({x}) → CH(X) → CH(Y) → 0

is a split injection (with the splitting given by the degree homomorphism CH(X) → CH(Spec F)), we have

Lemma 70.2. Let x be a singular rational point of a quadric X and let Y be as above. Then CH_{0}(X) is (the infinite cyclic group) generated by [x] and CH_{i}(X) is isomorphic to CH_{i−1}(Y) for all i > 0 .

Now assume that φ is nonzero and that X = X_{φ} has a nonsingular rational point x ∈ X (although X itself may not be smooth). The point x is given by a nonzero isotropic vector v ∈ V. The projective quadric Y, given by the restriction of φ on the orthogonal complement v^{⊥} of v in V, is a closed subscheme of X containing x. Since φ ≠ 0, the vector v lies outside of the radical of φ, so that v^{⊥} is a hyperplane of V. If φ restricted to v^{⊥} is zero, then Y is a projective space; otherwise x is a singular point of Y (and codim_{X} Y = 1). The difference X \ Y is isomorphic to an affine space (cf. proof of Proposition 22.9). Thus we can apply Theorem 66.2 to X with the one term filtration Y ⊂ X. Theorem 66.2 and Lemma 70.2 yield (we also include the trivial case of φ = 0):

Lemma 70.3. Let x be a nonsingular rational point of a quadric X and Y be as above. Then the push-forward homomorphism CH_{i}(Y) → CH_{i}(X) is an isomorphism for any i < dim X. In particular, if dim X > 0, the group CH_{0}(X) is generated by [x].

Combining Lemma 70.2 with Lemma 70.3 and adding the trivial case of φ = 0 yields

Corollary 70.4. Let X be an isotropic (not necessarily smooth) projective quadric X. Then the degree homomorphism deg : CH_{0}(X) → ℤ is an isomorphism unless φ is a hyperbolic plane, i.e., X is a disjoint union of two copies of Spec F.

71. The Chow group of dimension 0 cycles on quadrics

Recall that for every p ∈ [0, n], the group CH^{p}(ℙ_{F}^{n}) is infinite cyclic generated by the class h^{p} where h ∈ CH^{1}(ℙ_{F}^{n}) is the class of a hyperplane in ℙ_{F}^{n} (cf. Example 57.23). Thus for every p ∈ [0, n] and α ∈ CH^{p}(ℙ_{F}^{n}), we have α = mh^{p} for a uniquely determined integer m. We call m the degree of α and write m = deg(α). We have deg(αβ) = deg(α) deg(β) for all homogeneous cycles α ∈ CH^{p}(ℙ_{F}^{n}) and β ∈ CH^{q}(ℙ_{F}^{n}) satisfying p, q ≥ 0 and p + q ≤ n.

If Z is a closed subvariety of ℙ_{F}^{n}, we define the degree deg(Z) of Z as deg([Z]).

Lemma 71.1. Let x ∈ ℙ_{F}^{n} be a closed point of degree d > 1 such that the field extension F (x)/F is simple (i.e., generated by one element). Then there is a morphism f : ℙ^{1} → ℙ^{n} with image a curve C satisfying x ∈ C and deg(C) < d.

Proof. Let u be a generator of the field extension F (x)/F. We can write the homogeneous coordinates s_{i} of x in the form s_{i} = f_{i}(u), i ∈ [0, n], where f_{i} are polynomials over F of degree less than d. Let k be the largest degree of the f_{i} and set F_{i}(t_{0}, t_{1}) = t_{1}^{k}f_{i}(t_{0}/t_{1}). The polynomials F_{i} are all homogeneous of degree k < d. We may assume that all the F_{i} are relatively prime (by dividing out the gcd of the F_{i}). Consider the morphism f : ℙ_{F}^{1} → ℙ_{F}^{n} given by the polynomials F_{i} and let C be the image of f. Note that C contains x and C(F) ≠ ∅. In particular, the map f is not constant. Therefore C is a closed curve in ℙ_{F}^{n}. We have f_{∗}([ℙ^{1}]) = r[C] for some r ≥ 1.

Choose an index i such that F_{i} is a nonzero polynomial and consider the hyperplane H in ℙ_{F}^{n} given by s_{i} = 0. The subscheme f^{−1}(H) ⊂ ℙ_{F}^{1} is given by F_{i}(t_{0}, t_{1}) = 0, so f^{−1}(H) is a 0-dimensional subscheme of degree k = deg F_{i}. Hence H has a proper inverse image with respect to f. By Proposition 57.19, we have f^{∗}(h) = mp, where p is the class of a point in ℙ_{F}^{1} and 1 ≤ m ≤ k < d. It follows from Proposition 56.9 that

h · r[C] = h · f_{∗}([ℙ^{1}]) = f_{∗}(f^{∗}(h)) = f_{∗}(mp) = mh^{n}.

Hence deg(C) = m/r ≤ m < d. □

We follow [66, Th. 3.2] in the proof of the next result.

Theorem 71.2. Let X be an anisotropic (not necessarily smooth) quadric over F and let x_{0} ∈ X be a closed point of degree 2. Then for every closed point x ∈ X, we have [x] = a[x_{0}] in CH_{0}(X) for some a ∈ ℤ.

Proof. We induct on d = deg x. Suppose first that there are no intermediate fields between F and F (x). In particular, the field extension F (x)/F is simple. The quadric X is a hypersurface in the projective space ℙ_{F}^{n} for some n. By Lemma 71.1, there is an integral closed curve C ⊂ ℙ_{F}^{n} of degree less than d with C(F) ≠ ∅ and x ∈ C.

Let g : X → ℙ_{F}^{n} be the closed embedding. Since X is anisotropic and C(F) ≠ ∅, C is not contained in X. Therefore, C has proper inverse image with respect to g. As x ∈ C ∩ X, by Proposition 57.19,

g^{⋆}([C]) = [x] + α

in CH_{0}(X), where α is a nonnegative 0-dimensional cycle on X. By Proposition 56.11, we have

deg(g^{⋆}([C])) = deg(g_{∗} ∘ g^{⋆}([C])) = deg([X] · [C]) = 2 deg(C),

hence

deg α = 2 deg C − deg x = 2 deg C − d < d.

Thus the cycle α is supported on closed points of degree less than d. By the induction hypothesis, α = b[x_{0}] in CH_{0}(X) for some b ∈ ℤ. We also have [C] = c[L] in CH_{0}(ℙ_{F}^{n}) where L is a line in ℙ_{F}^{n} satisfying x_{0} ∈ L and c = deg C. Since L ∩ X = {x_{0}}, by Corollary 57.20, we have g^{⋆}([L]) = [x_{0}]. Therefore,

[x] = g^{⋆}([C]) − α = cg^{⋆}([L]) − b[x_{0}] = (c − b)[x_{0}].

Now suppose that there is a proper intermediate field L between F and E = F (x). Let f denote the natural morphism X_{L} → X. The morphism Spec E → X induced by x and the inclusion of L into E define a closed point x′ ∈ X_{L} with f(x′) = x and F (x′) = E. It follows that f_{∗}([x′]) = [x].

Consider two cases:

Case 1: X_{L} is isotropic. Let y ∈ X_{L} be a rational point. Since deg f_{∗}([y]) = [L : F] < d, by the induction hypothesis, f_{∗}([y]) ∈ ℤ · [x_{0}]. By Corollary 70.4, we know that CH_{0}(X_{L}) is generated by the classes of rational points, hence [x] = f_{∗}([x′]) ∈ ℤ · [x_{0}].

Case 2: X_{L} is anisotropic. Applying the induction hypothesis to the quadric X_{L} and the point x′ of degree [E : L] < d, we have [x′] = b[(x_{0})_{L}] for some b ∈ ℤ. Hence

[x] = f_{∗}([x′]) = bc[x_{0}],

where c = [L : F]. □

We therefore obtain another proof of Springer’s Theorem 18.5.

Corollary 71.3 (Springer’s Theorem). If X is an anisotropic quadric, the image of the degree homomorphism deg : CH_{0}(X) → ℤ is equal to 2ℤ, i.e., the degree of a finite field extension L/F with X_{L} isotropic, is even.

In the case char F ≠ 2, the following important statement was proven in [69, Prop. 2.6] and by Swan in [130]. The case of characteristic 2 was considered by Totaro in [132].

Corollary 71.4. For every anisotropic quadric X, the degree homomorphism deg : CH_{0}(X) → ℤ is injective.

72. The reduced Chow group

Let X be an arbitrary smooth projective quadric. We write CH(X̄^{r}) for CH(X_{E}^{r}), where E is a field extension of F such that the quadric X_{E} is split. Note that for any field L containing E, the change of field homomorphism CH(X_{E}^{r}) → CH(X_{L}^{r}) of Example 49.14 is an isomorphism; therefore for any field extension E′/F with split X_{E′}, the groups CH(X_{E}^{r}) and CH(X_{E′}^{r}) are canonically isomorphic, hence CH(X̄^{r}) can be defined invariantly as the colimit of the groups CH(X_{L}^{r}), where L runs over all field extensions of F.

If D is even, an orientation of the quadric is the choice of one of two classes of d-dimensional linear subspaces in CH(X̄). An even-dimensional quadric with an orientation is called oriented.

The reduced Chow group \overline{CH}(X^{r}) is defined as the image of the change of field homomorphism CH(X^{r}) → CH(X̄^{r}).

We say that an element of CH(X̄^{r}) is rational if it lies in the subgroup \overline{CH}(X^{r}) ⊂ CH(X̄^{r}). More generally, for a field extension L/F, the elements of the subgroup \overline{CH}(X_{L}^{r}) ⊂ CH(X̄^{r}) are called L-rational.

Replacing the integral Chow group by the Chow group modulo 2 in the above definitions, we get the modulo 2 reduced Chow group \overline{Ch}(X^{r}) ⊂ Ch(X̄^{r}) and the corresponding notion of (L-)rational cycles modulo 2.

Abusing terminology, we shall often call elements of a Chow group, cycles. The basis described in Proposition 68.5 will be called the basis for CH(X̄^{r}) and its elements basis elements or basic cycles. Similarly, this basis modulo 2 will be called the basis for Ch(X̄^{r}) and its elements basis elements or basic cycles. We use the same notation for the basis elements of CH(X̄^{r}) and for their reductions modulo 2. The decomposition of an element α ∈ Ch(X̄^{r}) will always mean its representation as a sum of basic cycles. We say that a basis cycle β is contained in the decomposition of α (or simply “is contained in α”), if β is a summand of the decomposition. More generally, for two cycles α′, α ∈ Ch(X̄^{r}), we say that α′ is contained in α or that α′ is a subcycle of α (notation: α′ ⊂ α), if every basis element contained in α′ is also contained in α.

A basis element of Ch(X̄^{r}) is called nonessential, if it is an external product of (internal) powers of h (including h^{0} = 1 = [X̄]); the other basis elements are called essential. An element of Ch(X̄^{r}) that is a sum of nonessential basis elements, is called nonessential as well. Note that all nonessential elements are rational since h is rational. An element of Ch(X̄^{r}) that is a sum of essential basis elements, is called essential as well. (The zero cycle is the only element which is essential and nonessential simultaneously). The group Ch(X̄^{r}) is a direct sum of the subgroup of nonessential elements and the subgroup of essential elements. We call the essential component of an element α ∈ Ch(X̄^{r}) the essence of α. Clearly, the essence of a rational element is rational.

The group \overline{Ch}(X) is easy to compute. First of all, by Springer’s theorem (Corollary 71.3), one has

Lemma 72.1. If the quadric X is anisotropic (i.e., X(F) = ∅), then the element l_{0} ∈ Ch(X̄) is not rational.

Corollary 72.2. If X is anisotropic, the group \overline{Ch}(X) is generated by the nonessential basis elements.

Proof. If the decomposition of an element α ∈ \overline{Ch}(X) contains an essential basis element l_{i} for some i ≠ D/2, then l_{i} ∈ \overline{Ch}(X) because l_{i} is the i-dimensional homogeneous component of α (and \overline{Ch}(X) is a graded subring of Ch(X̄)). If the decomposition of an element α ∈ \overline{Ch}(X) contains the essential basis element l_{i} for i = D/2, then D/2 = d, and the d-dimensional homogeneous component of α is either l_{d} or l_{d} + h^{d} so we still have l_{i} ∈ \overline{Ch}(X). It follows that l_{0} = l_{i} · h^{i} ∈ \overline{Ch}(X), contradicting Lemma 72.1. □

Let V be the underlying vector space of φ and let W ⊂ V be a totally isotropic subspace of dimension a ≤ d. Let Y be the projective quadric of the quadratic form ψ : W^{⊥}/W → F induced by φ. Then ψ is nondegenerate, Witt-equivalent to φ, dim ψ = dim φ − 2a, and dim Y = dim X − 2a. A point of the product Y × X is a pair (A/W, B), where B ⊂ V is a totally isotropic subspace of dimension 1 and A ⊂ W^{⊥} is a totally isotropic subspace of dimension a + 1 containing W. Let Z ⊂ Y × X be the closed scheme of the pairs (A/W, B) satisfying the condition B ⊂ A. Note that the composition Z ↪ Y × X ⟶^{pr_{Y}} Y is an a-dimensional projective bundle; in particular, Z is equidimensional (and Z is a variety if Y is) of dimension dim Z = dim Y + a = dim X − a. Its class α = [Z] ∈ CH(Y × X) is called the incidence correspondence.

We first note that the inverse image pr_{X}^{−1}(ℙ(W)) of the closed subvariety ℙ(W) ⊂ X under the projection pr_{X} : Y × X → X is contained in Z with complement a dense open subscheme of Z mapping under pr_{X} isomorphically onto (ℙ(W^{⊥}) ∩ X) \ ℙ(W).

We let h^{i} = 0 = l_{i} for any negative integer i.

Lemma 72.3. Let α be the incidence correspondence in CH(Y × X).

(1) For any i = 0, …, d − a, the homomorphism α_{∗} : CH(Ȳ) → CH(X̄) takes

h^{i} to h^{i+a} and l_{i} to l_{i+a}.

(2) For any i = 0, …, d, the homomorphism α^{∗} : CH(X̄) → CH(Ȳ) takes h^{i} to h^{i−a} and l_{i} to l_{i−a}. (In the case of even D, the two formulas involving l_{d} are true for an appropriate choice of orientations of X and of Y.)

Proof. For an arbitrary i ∈ [0, d − a], let L ⊂ W^{⊥}/W be a totally isotropic linear subspace of dimension i + 1. Then l_{i} = [ℙ(L)] ∈ CH(Ȳ). Let p : W^{⊥} → W^{⊥}/W be the projection. Since the dense open subscheme ( pr_{Y}^{−1}(ℙ(L)) ∩ Z) \ pr_{X}^{−1}(ℙ(W)) of the intersection pr_{Y}^{−1}(ℙ(L)) ∩ Z maps under pr_{X} isomorphically onto ℙ(p^{−1}(L)) \ ℙ(W), we have (using Proposition 57.21): α_{∗}(l_{i}) = [ℙ(p^{−1}(L))] = l_{i+a} ∈ CH(X̄). Similarly, for any linear subspace H ⊂ W^{⊥}/W of codimension i, the element h^{i} ∈ CH(Ȳ) is the class of the intersection ℙ(H) ∩ Y and maps under α_{∗} to the class of [ℙ(p^{−1}(H)) ∩ X] which equals h^{i+a}.

To prove the statements on α^{∗} for an arbitrary i ∈ [a, d], let L ⊂ V be an (i + 1)-dimensional totally isotropic subspace satisfying dim(L ∩ W^{⊥}) = dim L − a and L ∩ W = 0. (The second condition is, in fact, a consequence of the first one.) Then l_{i} = [ℙ(L)] ∈ CH(X̄) and the intersection pr_{X}^{−1}(ℙ(L)) ∩ Z maps under pr_{Y} isomorphically onto ℙ(((L∩W^{⊥})+W)/W ); consequently, α^{∗}(l_{i}) = l_{i−a}. Similarly, if H ⊂ V is a linear subspace of codimension i satisfying dim(H ∩W^{⊥}) = dim H − a and H ∩ W = 0, then h^{i} = [ℙ(H) ∩ X] ∈ CH(X̄) and the intersection pr_{X}^{−1} (ℙ(H)∩X)∩Z maps under pr_{Y} isomorphically onto ℙ(((H ∩W^{⊥})+W)/W )∩ Y; consequently, α^{∗}(h^{i}) = h^{i−a}. □

Corollary 72.4. Assume that X is isotropic but not split and set a = 𝔦_{0}(X). Let X_{0} be the projective quadric given by an anisotropic quadratic form Witt-equivalent to φ (so dim X_{0} = D − 2a). Then

(1) The group Ch_{D−a}(X × X_{0}) contains a correspondence “pr” such that the induced homomorphism pr_{∗} : Ch(X̄) → Ch(X̄_{0}) takes h^{i} to h^{i−a} and l_{i} to l_{i−a} for i = 0, …, d.

(2) The group Ch_{D−a}(X_{0} × X) contains a correspondence “in” such that the induced homomorphism in_{∗} : Ch(X̄_{0}) → Ch(X̄) takes h^{i} to h^{i+a} and l_{i} to l_{i+a} for i = 0, …, d − a.

Remark 72.5. Note that the homomorphisms in_{∗} and pr_{∗} of Corollary 72.4 map rational cycles to rational cycles. Since the composite pr_{∗} ∘ in_{∗} is the identity, it follows that pr_{∗}(\overline{Ch}(X)) = \overline{Ch}(X_{0}). More generally, for any r ≥ 1 the homomorphisms

in_{∗}^{r} : Ch(X̄_{0}^{r}) → Ch(X̄^{r}) and pr_{∗}^{r} : Ch(X̄^{r}) → Ch(X̄_{0}^{r}),

induced by the rth tensor powers in^{r} ∈ Ch(X_{0}^{r} × X^{r}) and pr^{r} ∈ Ch(X^{r} × X_{0}^{r}) of the correspondences in and pr, map rational cycles to rational cycles and satisfy the relations pr_{∗}^{r} ∘ in_{∗}^{r} = id and pr_{∗}^{r}(\overline{Ch}(X^{r})) = \overline{Ch}(X_{0}^{r}).

We obtain now the following extension of Lemma 72.1.

Corollary 72.6. Let X be an arbitrary quadric and let i be any integer. Then l_{i} ∈ \overline{Ch}(X) if and only if 𝔦_{0}(X) > i.

Proof. The “if” part of the statement is trivial. We prove the “only if” part by induction on i. The case i = 0 is Lemma 72.1.

We may assume that i > 0 and l_{i} ∈ \overline{Ch}(X). Since l_{i} · h = l_{i−1}, the element l_{i−1} is also rational. Therefore 𝔦_{0}(X) ≥ i by the induction hypothesis. If 𝔦_{0}(X) = i the image of l_{i} ∈ \overline{Ch}(X) under the map pr_{∗} : Ch(X̄) → Ch(X̄_{0}) of Corollary 72.4 equals l_{0} and is rational. Therefore, by Lemma 72.1, the quadric X_{0} is isotropic, a contradiction. □

The following observation is crucial:

Theorem 72.7. The absolute and relative higher Witt indices of a nondegenerate quadratic form φ are determined by the group

\overline{Ch}(X^{∗}) = ∐_{r≥1} \overline{Ch}(X^{r}).

Proof. We first note that the group \overline{Ch}(X) determines 𝔦_{0}(φ) by Corollary 72.6.

Let X_{0} be as in Corollary 72.4. By Corollary 72.4 and Remark 72.5, the group \overline{Ch}(X_{0}^{∗}) is recovered as the image of the group \overline{Ch}(X^{∗}) under the homomorphism Ch(X̄^{∗}) → Ch(X̄_{0}^{∗}) induced by the tensor powers of the correspondence pr.

Let F_{1} be the first field in the generic splitting tower of φ. The pull-back homomorphism g_{1}^{∗} : Ch(X_{0}^{r}) → Ch((X_{0})_{F_{1}}^{r−1}) with respect to the morphism of schemes g_{1} : (X_{0})_{F_{1}}^{r−1} → X_{0}^{r} given by the generic point of the first factor of X_{0}^{r}, is surjective (cf. Corollary 57.11). It induces an epimorphism \overline{Ch}(X_{0}^{r}) ↠ \overline{Ch}((X_{0})_{F_{1}}^{r−1}), which is the restriction of the epimorphism Ch(X̄_{0}^{r}) ↠ Ch(X̄_{0}^{r−1}) mapping each basis element of the form h^{0} ×β, β ∈ Ch(X̄_{0}^{r−1}), to β and killing all other basis elements. Therefore, the group \overline{Ch}(X_{0}^{∗}) determines the group \overline{Ch}((X_{0})_{F_{1}}^{∗}), and we finish by induction on the height 𝔥(φ). □

Remark 72.8. The proof of Theorem 72.7 shows that the statement of Theorem 72.7 can be made more precise in the following way: If for some q ∈ [0, 𝔥(φ)] the absolute Witt indices 𝔧_{0}, …, 𝔧_{q−1} are already known, then one determines 𝔧_{q} by the formula

𝔧_{q} = max {j | the product h^{𝔧_{0}} × h^{𝔧_{1}} × ⋯ × h^{𝔧_{q−1}} × l_{j−1}

is contained in a rational cycle}.

73. Cycles on X^{2}

In this section, we study the groups \overline{Ch}_{i}(X^{2}) for i ≥ D. After Lemma 73.2, we shall assume that X is anisotropic.

Most results of this section are simplified versions of original results on integral motives of quadrics due to Vishik in [133].

Lemma 73.1. The sum

Δ = ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}) ∈ Ch(X̄^{2})

is always rational.

Proof. Either the composition with correspondence Δ or the composition with the correspondence Δ + h^{d} × h^{d} (depending on whether l_{d}^{2} is zero or not) induces the identity endomorphism of Ch(X̄^{2}). Therefore, this correspondence is the class of the diagonal which is rational. □

Lemma 73.2. Suppose for some i ∈ [1, d] at least one of the basis elements l_{d} × l_{i} or l_{i} × l_{d} of the group Ch(X̄^{2}) appears in the decomposition of a rational cycle. Then X is hyperbolic.

73. CYCLES ON X^{2} 317

Proof. Let α be a cycle in \overline{Ch}_{i+d}(X^{2}) containing l_{i} × l_{d} or l_{d} × l_{i}. Replacing α by its transpose if necessary we may assume that l_{i} × l_{d} ∈ α. If β ∈ CH_{i+d}(X̄^{2}) is a basic cycle, then

β_{∗}(h^{i}) = \{ \begin{matrix}l_{d} & if β = l_{i} × l_{d}, \\ h^{d} & if β = l_{i} × h^{d}, \\ 0 & otherwise, \end{matrix}

hence α_{∗}(h^{i}) is equal to l_{d} or h^{d} + l_{d}. As α_{∗}(h^{i}) is rational, the cycle l_{d} is rational and X is hyperbolic by Corollary 72.6. □

We assume now that X is anisotropic throughout the rest of this section. Let α_{1}, α_{2} ∈ \overline{Ch}_{∗}(X^{2}). The intersection α_{1} ∩ α_{2} denotes the sum of the basic

cycles contained simultaneously in α_{1} and α_{2}.

Lemma 73.3. If α_{1}, α_{2} ∈ ∐_{i≥0} \overline{Ch}_{D+i}(X^{2}), then the cycle α_{1} ∩ α_{2} is rational.

Proof. Clearly, we may assume that α_{1} and α_{2} are homogeneous of the same dimension D + i and do not contain any nonessential basis element. Using Lemma 73.2 we see that the intersection α_{1} ∩ α_{2} is the essence of the composite of rational correspondences α_{2} ∘ (α_{1} · (h^{0} × h^{i})) and hence is rational. □

Notation 73.4. We write \overline{Ch}e(X̄^{2}) for the group of essential elements in

∐_{i≥D} Ch_{i}(X̄^{2})

and \overline{Ch}e(X^{2}) for the group of rational elements in \overline{Ch}e(X̄^{2}).

Definition 73.5. A nonzero element of \overline{Ch}e(X^{2}) is called minimal if it does not contain any proper rational subcycle.

Note that a minimal cycle is always homogeneous.

Proposition 73.6. Let X be a smooth anisotropic quadric. Then (1) The minimal cycles form a basis of the group \overline{Ch}e(X^{2}).

(2) Two different minimal cycles intersect trivially.

(3) The sum of the minimal cycles of dimension D is equal to the sum

∑_{i=0}^{d} h^{i} × l_{i} + l_{i} × h^{i}

of all D-dimensional essential basis elements (excluding l_{d} × l_{d} in the case of even D).

Proof. The first two statements of Proposition 73.6 follow from Lemma 73.3. The last statement follows from the previous ones together with Lemma 73.1. □

Let α be an element of Ch_{D+r}(X̄^{2}) for some r ≥ 0. For every i ∈ [0, r], the products α · (h^{0} × h^{i}), α · (h^{1} × h^{i−1}), …, α · (h^{i} × h^{0}) will be called the (ith order) derivatives of α.

Note that all the derivatives of a rational cycle are also rational.

Lemma 73.7. (1) A derivative of an essential basis element β ∈ \overline{Ch}e_{D+r}(X̄^{2}) is an essential basis element.

(2) For any r ≥ 0, nonnegative integers i_{1}, j_{1}, i_{2}, j_{2} satisfying i_{1} + j_{1} ≤ r, i_{2} + j_{2} ≤ r, and nonzero essential cycle β ∈ \overline{Ch}e_{D+r}(X̄^{2}), the two derivatives β · (h^{i_{1}} × h^{j_{1}}) and β · (h^{i_{2}} × h^{j_{2}}) of β coincide only if i_{1} = i_{2} and j_{1} = j_{2}.

(3) For any r ≥ 0, nonnegative integers i, j with i+j ≤ r, and nonzero essential cycles β_{1}, β_{2} ∈ \overline{Ch}e_{D+r}(X̄^{2}), the derivatives β_{1} · (h^{i} × h^{j}) and β_{2} · (h^{i} × h^{j}) of β_{1} and β_{2} coincide only if β_{1} = β_{2}.

Proof. (1): If β is an essential basis element of \overline{Ch}e_{D+r}(X̄^{2}) for some r > 0, then up to transposition, β = h^{i} × l_{i+r} for some i ∈ [0, d − r]. An arbitrary derivative of β is equal to β · (h^{j_{1}} × h^{j_{2}}) = h^{i+j_{1}} × l_{i+r−j_{2}} for some j_{1}, j_{2} ≥ 0 such that j_{1} + j_{2} ≤ r. It follows that the integers i + j_{1} and i + r − j_{2} are in the interval [0, d]; therefore, h^{i+j_{1}} × l_{i+r−j_{2}} is an essential basis element.

Statement (2) and (3) are left to the reader. □

Remark 73.8. To visualize the above, it is convenient to think of the essential basic cycles in ∐_{i≥D} Ch_{i}(X̄^{2}) (with l_{D/2} ×l_{D/2} excluded by Lemma 73.2) as points of two “pyramids”. For example, if D = 8 or D = 9, we write

∗  ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

If we count the rows of the pyramids from the bottom starting with 0, the top row has number d, and for every r = 0, …, d, the rth row of the left pyramid represents the essential basis elements h^{i} × l_{r+i}, i = 0, 1, …, d − r of Ch_{D+r}(X̄^{2}), while the rth row of the right pyramid represents the essential basis elements l_{r+i} × h^{i}, i = d − r, d − r − 1, …, 0 (so that the basis elements of each row are ordered by the codimension of the first factor).

For any α ∈ Ch(X̄^{2}), we fill in the pyramids by putting a mark in the points representing basis elements contained in the decomposition of α; the picture thus obtained is the diagram of α. If α is homogeneous, the marked points (if any) lie in the same row. It is now easy to interpret the derivatives of α if α is homogeneous of dimension ≥ D: the diagram of an ith order derivative is a translation of the marked points of the diagram of α moving them i rows lower. In particular, the diagram of every derivative of such an α has the same number of marked points as the diagram of α (cf. Lemma 73.7). The diagrams of any two different derivatives of the same order are shifts (to the right or to the left) of each other.

Example 73.9. Let D = 8 or D = 9. Let α ∈ Ch_{D+1}(X̄^{2}) be the essential cycle α = h^{0} × l_{1} + h^{2} × l_{3} + l_{3} × h^{2}. Then the diagram of α is

∗  ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ • ∗ • ∗  ∗ • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

There are precisely two first-order derivatives of α. They are given by α·(h^{0}×h^{1}) = h^{0} × l_{0} + h^{2} × l_{2} + l_{3} × h^{3} and α · (h^{1} × h^{0}) = h^{1} × l_{1} + h^{3} × l_{3} + l_{2} × h^{2}. Their diagrams are as follows:

∗  ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ • ∗ • ∗ ∗ ∗ • ∗ ∗ ∗
∗  ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ • ∗ • ∗ ∗ ∗ • ∗ ∗

Lemma 73.10. Let α ∈ \overline{Ch}e(X^{2}). Then the following conditions are equivalent: (1) The cycle α is minimal.

(2) All derivatives of α are minimal.

(3) At least one derivative of α is minimal.

Proof. Derivatives of a proper subcycle of α are proper subcycles of the derivatives of α; therefore, (3) ⇒ (1).

In order to show that (1) ⇒ (2), it suffices to show that the two first order derivatives α·(h^{0}×h^{1}) and α·(h^{1}×h^{0}) of a minimal cycle α are also minimal. If not, possibly replacing α by its transposition, we reduce to the case where the derivative α · (h^{0} × h^{1}) of a minimal α is not minimal. It follows that the cycle α · (h^{0} × h^{i}) with i = dim α − D is also not minimal. Let α′ be its proper subcycle. Taking the essence of the composite α ∘ α′, we get a proper subcycle of α, a contradiction. □

Corollary 73.11. The derivatives of a minimal cycle are disjoint.

Proof. The derivatives of a minimal cycle are minimal by Lemma 73.10 and pairwise different by Lemma 73.7. As two different minimal cycles are disjoint by Lemma 73.3, the result follows. □

We recall the notation of §25. If F_{0} = F, F_{1}, …, F_{𝔥} is the generic splitting tower of φ with 𝔥 = 𝔥(φ) the height of φ, and φ_{i} = (φ_{F_{i}})_{an} for i ≥ 0, we let X_{i} = X_{φ_{i}} be the projective quadric over F_{i} given by φ_{i}. Then 𝔦_{r} = 𝔦_{r}(φ) = 𝔦_{r}(X) is the rth relative and 𝔧_{r} = 𝔧_{r}(φ) = 𝔧_{r}(X) the rth absolute higher Witt index of φ, r ∈ [0, 𝔥]. We will also call these numbers the relative and the absolute Witt indices of X respectively.

Lemma 73.12. If i, j are integers in the interval [0, d] satisfying i < 𝔧_{q} ≤ j for some q ∈ [1, 𝔥), then no element in \overline{Ch}(X^{2}) contains either h^{i} × l_{j} or l_{j} × h^{i}.

Proof. Let i, j be integers of the interval [0, d] such that h^{i} × l_{j} or l_{j} × h^{i} appears in the decomposition of some α ∈ \overline{Ch}(X^{2}). Replacing α by its transpose if necessary, we may assume that h^{i} × l_{j} ∈ α. Replacing α by its homogeneous component containing h^{i} × l_{j}, we reduce to the case that α is homogeneous.

Suppose q is an integer in [1, 𝔥) satisfying i < 𝔧_{q}. It suffices to show that j < 𝔧_{q} as well.

Let L be a field extension of F with 𝔦_{0}(X_{L}) = 𝔧_{q} (e.g., L = F_{q}). The cycles α and l_{i} are both L-rational. Therefore, so is the cycle α_{∗}(l_{i}) = l_{j}. It follows by Corollary 72.6 that j < 𝔧_{q}. □

Remark 73.13. In order to “see” the statement of Lemma 73.12, it is helpful to mark by a ∗ only those essential basis elements that are not “forbidden” by this lemma in the pyramids of basic cycles drawn in Remark 73.8 and to mark by a ∘ the remaining points of the pyramids. We will get isosceles triangles based on the lower row of these pyramids. For example, if X is a 34-dimensional quadric with relative higher Witt indices 4, 2, 4, 8, the picture looks as follows:

∘  ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗  ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗  ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗  ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∗ ∗ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∗ ∗ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∘ ∗ ∘ ∗ ∗ ∗ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∗ ∗ ∗ ∘ ∗ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

The triangles in the pyramids of Remark 73.13 will be called shell triangles. The shell triangles in the left pyramid are numbered from the left starting with 1. The shell triangles in the right pyramid are numbered from the right starting with 1 as well (so that the symmetric triangles have the same number; for any q ∈ [1, 𝔥], the bases of the qth triangles have (each) 𝔦_{q} points). The rows of the shell triangles are numbered from below starting with 0. The points of rows of the shell triangles (of the left ones as well as of the right ones) are numbered from the left starting with 1.

Lemma 73.14. For every rational cycle α ∈ ∐_{i≥D} \overline{Ch}_{i}(X^{2}), the number of essential basic cycles contained in α is even (i.e., the number of the marked points in the diagram of α is even).

Proof. We may assume that α is homogeneous, say, α ∈ \overline{Ch}_{D+k}(X^{2}), k ≥ 0. We may also assume that k ≤ d as there are no essential basic cycles of dimension > D + d. Let n be the number of essential basic cycles contained in α. The pullback δ^{∗}(α) of α with respect to the diagonal δ : X → X^{2} produces n · l_{k} ∈ \overline{Ch}(X). It follows by Corollary 72.2 that n is even. □

Lemma 73.15. Let α ∈ \overline{Ch}(X^{2}) be a cycle containing the top of a qth shell triangle for some q ∈ [1, 𝔥]. Then α also contains the top of the other qth shell triangle.

Proof. We may assume that α contains the top of the left qth shell triangle. Replacing F by the field F_{q−1}, X by X_{q−1}, and α by pr_{∗}^{2}(α), where pr ∈ Ch(X_{F_{q−1}} × X_{q−1}) is the correspondence of Corollary 72.4, we may assume that q = 1.

Replacing α by its homogeneous component containing the top of the left first shell triangle β = h^{0} × l_{𝔧_{1}−1}, we may assume that α is homogeneous.

Suppose that the transpose of β is not contained in α. By Lemma 73.12, the element α does not contain any essential basic cycles having h^{i} with 0 < i < 𝔦_{1} as a factor. Since α ≠ β by Lemma 73.14, we have 𝔥 > 1. Moreover, the number of essential basis elements contained in α and the number of essential basis elements contained in pr_{∗}^{2}(α) ∈ \overline{Ch}(X_{1}^{2}) differ by 1. In particular, these two numbers have different parity. However, the number of the essential basis elements contained in α is even by Lemma 73.14. By the same lemma, the number of essential basis elements contained in pr_{∗}^{2}(α) is even, too. □

Definition 73.16. A minimal cycle α ∈ \overline{Ch}e(X^{2}) is called primordial if it is not a derivative of positive order of another rational cycle.

Lemma 73.17. Let α ∈ \overline{Ch}e(X^{2}) be a minimal cycle (as defined in 73.5) which also contains the top of a qth shell triangle for some q ∈ [1, 𝔥]. Then α is symmetric and primordial.

Proof. The cycle α ∩ t(α), where t(α) is the transpose of α (where the intersection of cycles is defined in Lemma 73.3) is symmetric, rational by Lemma 73.3, contained in α, and, by Lemma 73.15, still contains the tops h^{𝔧_{q−1}} × l_{𝔧_{q}−1} and l_{𝔧_{q}−1} × h^{𝔧_{q−1}} of both qth shell triangles. Therefore, it coincides with α by the minimality of α.

It is easy to “see” that α is primordial by looking at the picture in Remark 73.13. Nevertheless, we prove it. If there exists a rational cycle β ≠ α such that α is a derivative of β, then there exists a rational cycle β′ such that α is an order one derivative of β′, i.e., α = β′ · (h^{0} × h^{1}) or α = β′ · (h^{1} × h^{0}). In the first case, β′ would contain the basic cycle h^{𝔧_{q−1}} × l_{𝔧_{q}} while in the second case β′ would contain h^{𝔧_{q−1}−1}×l_{𝔧_{q}−1}. However, neither of these two cases is possible by Lemma 73.12. □

It is easy to see that a cycle α satisfying the hypothesis of Lemma 73.17 with q = 1 exists:

Lemma 73.18. There exists a cycle in \overline{Ch}_{D+𝔦_{1}−1}(X^{2}) containing the top h^{0}×l_{𝔦_{1}−1} of the first left shell triangle.

Proof. If D = 0, this follows by Lemma 73.1. So assume D > 0. Consider the pull-back homomorphism \overline{Ch}(X^{2}) ↠ \overline{Ch}(X_{F (X)}) with respect to the morphism X_{F (X)} → X^{2} produced by the generic point of the first factor of X^{2}. By Corollary 57.11, this is an epimorphism. It is also a restriction of the homomorphism Ch(X̄^{2}) → Ch(X̄) mapping each basis element of the type h^{0}×l_{i} to l_{i} and vanishing on all other basis elements. Therefore an arbitrary preimage of l_{𝔦_{1}−1} ∈ \overline{Ch}(X_{F (X)}) under the surjection \overline{Ch}(X^{2}) ↠ \overline{Ch}(X_{F (X)}) contains h^{0} × l_{𝔦_{1}−1}. □

Lemma 73.19. Let ρ ∈ \overline{Ch}_{D}(X^{2}), q ∈ [1, 𝔥], and i ∈ [1, 𝔦_{q}]. Then the element h^{𝔧_{q−1}+i−1} × l_{𝔧_{q−1}+i−1} is contained in ρ if and only if the element l_{𝔧_{q}−i} × h^{𝔧_{q}−i} is contained in ρ.

Proof. Clearly, it suffices to prove Lemma 73.19 for q = 1. By Lemma 73.18, the basis element h^{0} × l_{𝔦_{1}−1} is contained in a rational cycle. Let α be the minimal cycle containing h^{0} × l_{𝔦_{1}−1}. By Lemma 73.15, the cycle α also contains l_{𝔦_{1}−1} × h^{0}. Therefore, the derivative α·(h^{i−1}×h^{𝔦_{1}−i}) contains both h^{i−1}×l_{i−1} and l_{𝔦_{1}−i}×h^{𝔦_{1}−i}. Since the derivative of a minimal cycle is minimal by Lemma 73.10, the lemma follows by Lemma 73.3. □

In the language of diagrams, the statement of Lemma 73.19 means that the ith point of the base of the qth left shell triangle in the diagram of ρ is marked if and only if the ith point of the base of the qth right shell triangle is marked.

Definition 73.20. The symmetric shell triangles (i.e., both qth shell triangles for some q) are called dual. Two points are called dual, if one of them is in a left shell triangle, while the other one is in the same row of the dual right shell triangle and has the same number as the first point.

Corollary 73.21. In the diagram of an element of \overline{Ch}(X^{2}), any two dual points are simultaneously marked or not marked.

Proof. Let k be the number of the row containing two given dual points. The case of k = 0 is treated in Lemma 73.19 (while Lemma 73.15 treats the case of “locally maximal” k). The case of an arbitrary k is reduced to the case of k = 0 by taking a kth order derivative of α. □

Remark 73.22. By Corollary 73.21, it follows that the diagram of a cycle in \overline{Ch}(X^{2}) is determined by one (left or right) half of itself. By “shell triangles” we shall mean the left shell triangles. Note also that the transposition of a cycle acts symmetrically about the vertical axis on each shell triangle.

The following proposition generalizes Lemma 73.18.

Proposition 73.23. Let f : \overline{Ch}(X^{2}) → [1, 𝔥] be the map that assigns to each γ ∈ \overline{Ch}(X^{2}) the integer q ∈ [1, 𝔥] such that the diagram of γ has a point in the qth shell triangle and has no points in the ith shell triangles for any i < q. For any q ∈ f( \overline{Ch}(X^{2})), there exists an element α ∈ \overline{Ch}(X^{2}) with f(α) = q so that α contains the top of the qth shell triangle.

Proof. We induct on q. If q = 1, the condition of Proposition 73.23 is automatically satisfied by Lemma 73.1 and the result follows by Lemma 73.18. So we may assume that q > 1.

Let γ be an element of \overline{Ch}(X^{2}) with f(γ) = q. Replacing γ by its appropriate homogeneous component, we may assume that γ is homogeneous. Replacing this homogeneous γ by any one of its maximal order derivatives, we may further assume that γ ∈ \overline{Ch}_{D}(X^{2}).

Let i be the smallest integer such that h^{𝔧_{q−1}+i} × l_{𝔧_{q−1}+i} ∈ γ. We first prove that the group \overline{Ch}(X^{2}) contains a cycle γ′ satisfying f(γ′) = q with γ′ containing h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1}. (This is the point on the right side of the qth shell triangle such that the line connecting it with h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1+i} is parallel to the left side of the shell triangle. If i = 0, then we can take α = γ′ and finish the proof.)

Let

pr_{∗}^{2} : \overline{Ch}(X_{F (X)}^{2}) → \overline{Ch}(X_{1}^{2}) and in_{∗}^{2} : \overline{Ch}(X_{1}^{2}) → \overline{Ch}(X_{F (X)}^{2})

be the homomorphisms of Remark 72.5. Applying the induction hypothesis to the quadric X_{1} and the cycle pr_{∗}^{2}(γ) ∈ \overline{Ch}(X_{1}^{2}), we get a homogeneous cycle in \overline{Ch}_{D+𝔦_{q}−1}(X_{F (X)}^{2}) containing h^{𝔧_{q−1}} × l_{𝔧_{q}−1}. Multiplying it by h^{i} × h^{0}, we get a homogeneous cycle in \overline{Ch}(X_{F (X)}^{2}) containing h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1}. Note that the quadric X_{F (X)} is not hyperbolic (since 𝔥 ≥ q > 1) and therefore, by Lemma 73.2, the basis element l_{d} × l_{d} is not contained in this cycle. Therefore, the group \overline{Ch}(X^{3}) contains a homogeneous cycle μ containing h^{0} × h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1} (and not containing h^{0} × l_{d} × l_{d}). View μ as a correspondence of the middle factor of X^{3} into the product of the two outer factors. Composing it with γ and taking the pull-back with respect to the partial diagonal map δ : X^{2} → X^{3}, (x_{1}, x_{2}) ↦ (x_{1}, x_{1}, x_{2}), we get the required cycle γ′ (more accurately, γ′ = δ^{∗}(t_{12}(μ) ∘ γ), where t_{12} is the automorphism of \overline{Ch}(X^{3}) given by the transposition of the first two factors of X^{3}).

The highest order derivative γ′ · (h^{𝔦_{q}−1−i} × h^{0}) of γ′ contains h^{𝔧_{q}−1} × l_{𝔧_{q}−1}, the last point of the base of the qth shell triangle. Therefore, the transpose t(γ′) contains the first point h^{𝔧_{q−1}} × l_{𝔧_{q−1}} of the base of the qth shell triangle by Remark 73.22. Replacing γ by t(γ′), we are in the case that i = 0 (see the third paragraph of the proof), finishing the proof. □

Illustration 73.24. The following picture shows the displacements of the special marked point of the qth shell triangle in the proof of Proposition 73.23:

6 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 5 ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ 4

We start with a cycle γ ∈ \overline{Ch}(X^{2}) with f(γ) = q. It contains a point somewhere in the qth shell triangle, the point in Position 1. Then we modify γ in such a way that f(γ) is always q and look at what happens with this point. Replacing γ by a maximal order derivative, we move the special point to the base of the shell triangle; for example, we can move it to Position 2. The heart of the proof is the movement from Position 2 to Position 3 (here we make use of the induction hypothesis). Again taking an appropriate derivative, we come to Position 4. Transposing this cycle, we come to Position 5. Finally, repeating the procedure used in the passage from Positions 2 to 3, we move from Position 5 to Position 6, arriving to the top.

Illustration 73.25. Let

\overline{Ch}(X^{2}) ↪ \overline{Ch}(X_{F (X)}^{2}) ⟶^{pr_{∗}^{2}} \overline{Ch}(X_{1}^{2})

be the homomorphism used in the proof of Proposition 73.23. If α ∈ \overline{Ch}(X^{2}), the diagram of pr_{∗}^{2}(α) is obtained from the diagram of α by erasing the first shell triangle. An example is shown by the illustration with X as in Remark 73.13:

∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗  ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗  ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗  ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗  ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗  ∘ ∘ ∗ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∗ ∗ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗  ∘ ∘ ∗ ∗ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ • ∗ • ∘ • ∘ • ∗ • ∘ • ∗ • ∗ • ∗ •  • ∘ • ∗ • ∘ • ∗ • ∗ • ∗ • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

diagram of α diagram of pr_{∗}^{2}(α)

Summarizing, we have the following structure result on \overline{Ch}e(X^{2}):

Theorem 73.26. Let X be a smooth anisotropic quadric. The set of primordial cycles Π lying in \overline{Ch}e(X^{2}) has the following properties:

(1) All derivatives of all cycles in Π are minimal and pairwise disjoint, and the set of these forms a basis of \overline{Ch}e(X^{2}). In particular, the sum of all maximal order derivatives of the elements in Π is equal to the cycle

Δ = ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}) ∈ Ch(X̄^{2}).

(2) Every cycle in Π is symmetric and has no points outside of the shell triangles.

(3) The map f in Proposition 73.23 is injective on Π. Every cycle π ∈ Π contains the top of the f(π)th shell triangle and has no points in any shell triangle with number in f(Π) \ {f(π)}.

(4) 1 ∈ f( \overline{Ch}(X^{2})) = f(Π).

Let f be as in Proposition 73.23. If f(α) = q for an element α ∈ Ch(X^{2}), we say that α starts in the qth shell triangle. More specifically, if f(π) = q for a primordial cycle π, we say that π is q-primordial.

The following statement is an additional property of 1-primordial cycles:

Proposition 73.27. Let π ∈ \overline{Ch}(X^{2}) be a 1-primordial cycle. Suppose π contains h^{i} × l_{i+𝔦_{1}−1} for some positive i ≤ d. Then the smallest integer i with this property coincides with an absolute Witt index of φ, i.e., i = 𝔧_{q−1} for some q ∈ [2, 𝔥].

Proof. The cycle π contains h^{0} × l_{𝔦_{1}−1} as this is the top of the first shell triangle. By Lemma 73.12, π contains none of the cycles h^{1} × l_{𝔦_{1}}, …, h^{𝔦_{1}−1} × l_{2𝔦_{1}−2}. It follows that if i ∈ [1, d] is the smallest integer satisfying h^{i} × l_{i+𝔦_{1}−1} ∈ π, then i ≥ 𝔧_{1} = 𝔦_{1}. Let q ∈ [2, 𝔥] be the largest integer with 𝔧_{q−1} ≤ i. We show that 𝔧_{q−1} = i. Suppose to the contrary that 𝔧_{q−1} < i.

Let X_{1} be the quadric over F (X) given by the anisotropic part of φ_{F (X)} and

pr_{∗}^{2} : \overline{Ch}(X_{F (X)}^{2}) → \overline{Ch}(X_{1}^{2}),

the homomorphism of Remark 72.5. Then the element pr_{∗}^{2}(π) starts in the shell triangle number q − 1 of X_{1}. Therefore, by Proposition 73.23, the quadric X_{1} possesses a (q − 1)-primordial cycle τ.

Let

in_{∗}^{2} : \overline{Ch}(X_{1}^{2}) → \overline{Ch}(X_{F (X)}^{2})

be the homomorphism of Remark 72.5. Then the cycle β = in_{∗}^{2}(τ) in \overline{Ch}(X_{F (X)}^{2}) contains h^{𝔧_{q−1}} × l_{𝔧_{q}−1} and does not contain any h^{j} × l_{?} with j < 𝔧_{q−1}.

Let η ∈ \overline{Ch}(X^{3}) be a preimage of β under the pull-back epimorphism

g^{∗} : \overline{Ch}(X^{3}) ↠ \overline{Ch}(X_{F (X)}^{2}) ,

where the morphism g : X_{F (X)}^{2} → X^{3} is induced by the generic point of the first factor of X^{3} (cf. Corollary 57.11). The cycle η contains h^{0} × h^{𝔧_{q−1}} × l_{𝔧_{q}−1} and does not contain any h^{0} × h^{j} × l_{?} with j < 𝔧_{q−1}.

We consider η as a correspondence X ⇝ X^{2}. Define μ as the composition μ = η ∘ α with α = π · (h^{0} × h^{𝔦_{1}−1}). The cycle α contains h^{0} × l_{0} and does not contain any h^{j} × l_{j} with j ∈ [1, i). In particular, since 𝔧_{q−1} < i, it does not contain any h^{j} × l_{j} with j ∈ [1, 𝔧_{q−1}]. Consequently, the cycle μ contains the basis element

h^{0} × h^{𝔧_{q−1}} × l_{𝔧_{q}−1} = (h^{0} × h^{𝔧_{q−1}} × l_{𝔧_{q}−1}) ∘ (h^{0} × l_{0})

and does not contain any h^{j} × h^{?} × l_{?} with j ∈ [1, 𝔧_{q−1}].

Let

δ^{∗} : \overline{Ch}(X^{3}) → \overline{Ch}(X^{2})

be the pull-back homomorphism with respect to the partial diagonal morphism

δ : X^{2} → X^{3}, (x_{1} × x_{2}) ↦ (x_{1} × x_{1} × x_{2}) .

The cycle δ^{∗}(μ) ∈ \overline{Ch}(X^{2}), contains the basis element

h^{𝔧_{q−1}} × l_{𝔧_{q}−1} = δ^{∗}(h^{0} × h^{𝔧_{q−1}} × l_{𝔧_{q}−1})

and does not contain any h^{j} × l_{?} with j < 𝔧_{q−1}. It follows that an appropriate derivative of the cycle δ^{∗}(μ) contains h^{i} × l_{i+𝔦_{1}−1} ∈ π and does not contain h^{0} × l_{𝔦_{1}−1} ∈ π. This contradicts the minimality of π. □

Remark 73.28. In the language of diagrams, Proposition 73.27 asserts that the point h^{i} × l_{i+𝔦_{1}−1} lies on the left side of the qth shell triangle.

Definition 73.29. We say that the integer q ∈ [2, 𝔥] occurring in Proposition 73.27 is produced by the 1-primordial cycle π.

CHAPTER XIV The Izhboldin Dimension

Let X be an anisotropic smooth projective quadric over a field F (of arbitrary characteristic). The Izhboldin dimension dim_{Izh} X of X is defined as

dim_{Izh} X := dim X − 𝔦_{1}(X) + 1 ,

where 𝔦_{1}(X) is the first Witt index of X.

Let Y be a complete (possibly singular) algebraic variety over F with all of its closed points of even degree and such that Y has a closed point of odd degree over F (X). The main theorem of this chapter is Theorem 76.1 below. It states that dim_{Izh} X ≤ dim Y and if dim_{Izh} X = dim Y the quadric X is isotropic over F (Y).

An application of Theorem 76.1 is the positive solution of a conjecture of Izhboldin that states: if an anisotropic quadric Y becomes isotropic over F (X), then dim_{Izh} X ≤ dim_{Izh} Y with equality if and only if X is isotropic over F (Y).

The results of this chapter for the case that characteristic ≠ 2 were obtained in [76].

74. The first Witt index of subforms

For the reader’s convenience, we list some easy properties of the first Witt index:

Lemma 74.1. Let φ be an anisotropic nondegenerate quadratic form over F of dimension at least two.

(1) The first Witt index 𝔦_{1}(φ) coincides with the minimal Witt index of φ_{E}, where E runs over all field extensions of F such that the form φ_{E} is isotropic.

(2) Let ψ be a nondegenerate subform of φ of codimension r and let E/F be a field extension. Then 𝔦_{0}(ψ_{E}) ≥ 𝔦_{0}(φ_{E})−r. In particular, 𝔦_{1}(ψ) ≥ 𝔦_{1}(φ)−r (if 𝔦_{1}(ψ) is defined, i.e., dim ψ ≥ 2).

Proof. The first statement is proven in Corollary 25.3. For the second statement, note that the intersection of a maximal isotropic subspace U (of dimension 𝔦_{0}(φ_{E})) of the form φ_{E} with the space of the subform ψ_{E} is of codimension at most r in U. □

The following two statements are due to Vishik (in the case that characteristic ≠ 2; cf. [133, Cor. 4.9]).

Proposition 74.2. Let φ be an anisotropic nondegenerate quadratic form over F with dim φ ≥ 2. Let ψ be a nondegenerate subform of φ. If codim_{φ} ψ ≥ 𝔦_{1}(φ), then the form ψ_{F (φ)} is anisotropic.

Proof. Let n = codim_{φ} ψ and assume that n ≥ 𝔦_{1}(φ). If the form ψ_{F (φ)} is isotropic, there exists a rational morphism X −−→ Y, where X and Y are the projective quadrics of φ and ψ respectively. We use the same notation as in §72. Let α ∈ \overline{Ch}(X^{2}) be the class of the closure of the graph of the composition X −−→ Y ↪ X. Since the push-forward of α with respect to the first projection X^{2} → X is nonzero, we have h^{0} × l_{0} ∈ α. On the other hand, as α is in the image of the push-forward homomorphism Ch(X̄ × Ȳ) → Ch(X̄^{2}) that maps any external product β × γ to β × in_{∗}(γ), where the push-forward in_{∗} : Ch(Ȳ) → Ch(X̄) maps h^{i} to h^{i+n}, and n ≥ 𝔦_{1}(φ), one has l_{𝔦_{1}(φ)−1} × h^{𝔦_{1}(φ)−1} ∉ α, contradicting Lemma 73.19 (cf. also Corollary 73.21). □

Corollary 74.3. Let φ be an anisotropic nondegenerate quadratic form and let φ′ be a nondegenerate subform of φ of codimension n with dim φ′ ≥ 2. If n < 𝔦_{1}(φ), then 𝔦_{1}(φ′) = 𝔦_{1}(φ) − n.

Proof. Let 𝔦_{1} = 𝔦_{1}(φ). By Lemma 74.1, we know that 𝔦_{1}(φ′) ≥ 𝔦_{1} − n. Let ψ be a nondegenerate subform of φ′ of dimension dim φ − 𝔦_{1}. If 𝔦_{1}(φ′) > 𝔦_{1} − n, then the form ψ_{F (φ)} is isotropic by Lemma 74.1 contradicting Proposition 74.2. □

Lemma 74.4. Let φ be an anisotropic nondegenerate quadratic F -form satisfying dim φ ≥ 3 and let 𝔦_{1}(φ) = 1. Let F (t)/F be a purely transcendental field extension of degree 1. Then there exists a nondegenerate subform ψ of φ_{F (t)} of codimension 1 satisfying 𝔦_{1}(ψ) = 1.

Proof. First consider the case that char(F) ≠ 2. We can write φ ≃ φ′⊥ ⟨a, b⟩ for some a, b ∈ F^{×} and some quadratic form φ′. Set

ψ = φ′_{F (t)}⊥ ⟨a + bt^{2}⟩ .

This is clearly a subform of φ_{F (t)} of codimension 1. Moreover, the fields F (t)(ψ) and F (φ) are isomorphic over F. In particular,

𝔦_{1}(ψ) = 𝔦_{0}(ψ_{F (t)(ψ)}) ≤ 𝔦_{0}(φ_{F (t)(ψ)}) = 𝔦_{0}(φ_{F (φ)}) = 𝔦_{1}(φ) = 1,

hence 𝔦_{1}(ψ) = 1.

Suppose char(F) = 2. If dim φ is even then φ ≃ φ′⊥[a, b] for some a, b ∈ F and some even-dimensional nondegenerate quadratic form φ′ by Proposition 7.31. In this case set

ψ = φ′_{F (t)}⊥ ⟨a + t + bt^{2}⟩ .

If dim φ is odd, then φ ≃ φ′⊥[a, b]⊥ ⟨c⟩ for some c ∈ F^{×}, some a, b ∈ F, and some even-dimensional nondegenerate quadratic form φ′. In this case set

ψ = φ′_{F (t)}⊥[a, b + ct^{2}].

In either case, ψ is a nondegenerate subform of φ_{F (t)} of codimension 1 such that the fields F (t)(ψ) and F (φ) are F -isomorphic. Therefore the argument above shows that 𝔦_{1}(ψ) = 1. □

75. Correspondences

Let X and Y be schemes over a field F. Suppose that X is equidimensional and let d = dim X. Recall that a correspondence (of degree zero) α : X ⇝ Y from X to Y is an element α ∈ CH_{d}(X × Y) (cf. §62). A correspondence is called prime if it is represented by a prime cycle. Every correspondence is a linear combination of prime correspondences with integer coefficients.

Let α : X ⇝ Y be a correspondence. Assume that X is a variety and Y is complete. The projection morphism p : X × Y → X is proper, hence the pushforward homomorphism

p_{∗} : CH_{d}(X × Y) → CH_{d}(X) = ℤ · [X]

is defined (cf. Proposition 49.9). The integer mult(α) ∈ ℤ satisfying

p_{∗}(α) = mult(α) · [X]

is called the multiplicity of α. Clearly, mult(α + β) = mult(α) + mult(β) for any two correspondences α, β : X ⇝ Y.

A correspondence α : Spec F ⇝ Y is represented by a 0-cycle z on Y. Clearly, mult(α) = deg(z), where deg : CH_{0}(Y) → ℤ is the degree homomorphism defined in Example 57.7. More generally, we have the following statement.

Lemma 75.1. The composition

CH_{d}(X × Y) → CH_{0}(Y_{F (X)}) ⟶^{deg} ℤ,

where the first map is the pull-back homomorphism with respect to the natural flat morphism Y_{F (X)} → X × Y takes a correspondence α to mult(α).

Proof. The statement follows by Proposition 49.20 applied to the fiber product diagram

Y_{F (X)} ⟶ X × Y
↓  ↓
Spec F (X) ⟶ X
□

Lemma 75.2. Let Y be a complete scheme and let F̃/F be a purely transcendental field extension. Then

deg CH_{0}(Y) = deg CH_{0}(YF̃).

Proof. We may assume that F̃ is the function field of the affine line 𝔸^{1}. By Proposition 49.20 applied to the fiber product diagram

Y_{F (𝔸^{1})}  ⟶^{f} Y
↓  ↓
Spec F (𝔸^{1}) ⟶ Spec F

it suffices to show that the pull-back homomorphism f^{∗} : CH_{0}(Y) → CH_{0}(Y_{F (𝔸^{1})}) is surjective. We have f = p ∘ q where p : Y × 𝔸^{1} → Y is the projection and q : Y_{F (𝔸^{1})} → Y × 𝔸^{1} is a natural morphism. The pull-back homomorphism p^{∗} is an isomorphism by Theorem 57.13 and q^{∗} is surjective by Corollary 57.11. As f^{∗} = q^{∗} ∘ p^{∗} by Proposition 49.18, the map f^{∗} is surjective. □

Corollary 75.3. Let Y be a complete variety, X a smooth projective quadric, and X′ ⊂ X an arbitrary closed subvariety of X. Then

deg CH_{0}(Y_{F (X)}) ⊂ deg CH_{0}(Y_{F (X′)}).

Proof. Since F (X) is a subfield of F (X × X′), we have

deg CH_{0}(Y_{F (X)}) ⊂ deg CH_{0}(Y_{F (X×X′)}).

As the quadric X_{F (X′)} is isotropic, the field extension F (X × X′)/F (X′) is purely transcendental. Hence

deg CH_{0}(Y_{F (X×X′)}) = deg CH_{0}(Y_{F (X′)})

by Lemma 75.2. □

Let X and Y be varieties over F with dim X = d and let Z ⊂ X ×Y be a prime d-dimensional cycle of multiplicity r > 0. The generic point of Z defines a degree r closed point of the generic fiber Y_{F (X)} of the projection X × Y → X and vice versa. Hence there is a natural bijection of the following two sets for every r > 0:

(1) prime d-dimensional cycles on X × Y of multiplicity r,

(2) closed points of Y_{F (X)} of degree r.

A rational morphism X −−→ Y defines a prime correspondence X ⇝ Y of multiplicity 1 by taking the closure of its graph. Conversely, a prime cycle Z ⊂ X × Y of multiplicity 1 is birationally isomorphic to X. Therefore, the projection to Y defines a rational map X −−→ Z → Y. Hence there are natural bijections between the sets of:

(0) rational morphisms X −−→ Y,

(1) prime d-dimensional cycles on X × Y of multiplicity 1,

(2) rational points of Y_{F (X)}.

A prime correspondence X ⇝ Y of multiplicity r can be viewed as a “generically r-valued map” between X and Y.

Let α : X ⇝ Y be a correspondence between varieties of dimension d. We write α^{t} : Y ⇝ X for the transpose of α (cf. §62).

Theorem 75.4. Let X be an anisotropic smooth projective quadric with 𝔦_{1}(X) = 1. Let δ : X ⇝ X be a correspondence. Then mult(δ) ≡ mult(δ^{t}) (mod 2).

Proof. The coefficient of h^{0} × l_{0} in the decomposition of the class of δ in the modulo 2 reduced Chow group \overline{Ch}(X^{2}) is mult(δ) (mod 2) (taking into account Lemma 73.2 in the case when dim X = 0). Therefore, the theorem is a particular case of Corollary 73.21. (It is also a particular case of Lemma 73.19 and also of Lemma 73.15.)

We give another proof of Theorem 75.4. By Example 66.6, we have

CH_{d}(X^{2}) ≃ CH_{d}(X) ⊕ CH_{d−1}(Fl) ⊕ CH_{0}(X) ,

where Fl is the flag variety of pairs (L, P), where L and P are a totally isotropic line and plane, respectively, satisfying L ⊂ P. It suffices to check the formula of Theorem 75.4 for δ lying in the image of any of these three summands.

Since the embedding CH_{d}(X) ↪ CH_{d}(X^{2}) is given by the push-forward with respect to the diagonal map, its image is generated by the diagonal class for which the congruence clearly holds.

Since X is anisotropic, every element of CH_{0}(X) becomes divisible by 2 over an extension of F by Theorem 71.2 and Proposition 68.1. As multiplicity is not changed under a field extension homomorphism, we have mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 2) for any δ in the image of CH_{0}(X).

Since the embedding CH_{d−1}(Fl) ↪ CH_{d}(X^{2}) is produced by a correspondence Fl ⇝ X^{2} of degree one, the image of CH_{d−1}(Fl) is contained in the image of the push-forward CH_{d}(Fl ×X^{2}) → CH_{d}(X^{2}) with respect to the projection. Let δ ∈ CH_{d}(X^{2}). By Lemma 75.1, the multiplicity of δ and of δ^{t} is the degree of the image of δ under the pull-back homomorphism CH_{d}(X^{2}) → CH_{0}(X_{F (X)}), given by the generic point of the appropriately chosen factor of X^{2}. As 𝔦_{1}(X) = 1, the degree of any closed point on (Fl ×X)_{F (X)} is even by Corollary 71.3. Consequently mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 2) for any δ in the image of CH_{d−1}(Fl). □

Remark 75.5. The statement of Theorem 75.4 with X replaced by an anisotropic nonsmooth quadric (over a field of characteristic 2) was proved by Totaro in [132].

Corollary 75.6. Let X be as in Theorem 75.4. Then any rational endomorphism f : X −−→ X is dominant. In particular, the only point x ∈ X admitting an F -embedding F (x) ↪ F (X) is the generic point of X.

Proof. Let δ : X ⇝ X be the class of the closure of the graph of f. Then mult(δ) = 1. Therefore, the integer mult(δ^{t}) is odd by Theorem 75.4. In particular, mult(δ^{t}) ≠ 0; i.e., f is dominant. □

76. The main theorem

The main theorem of the chapter is

Theorem 76.1. Let X be an anisotropic smooth projective quadric over F and let Y be a complete variety over F such that every closed point of Y is of even degree. If there is a closed point in Y_{F (X)} of odd degree, then

(1) dim_{Izh} X ≤ dim Y.

(2) If dim_{Izh} X = dim Y, then X is isotropic over F (Y).

Proof. A closed point of Y over F (X) of odd degree gives rise to a prime correspondence α : X ⇝ Y of odd multiplicity. By Springer’s Theorem 71.3, to prove statement (2), it suffices to find a closed point of X_{F (Y)} of odd degree, and equivalently, to find a correspondence Y ⇝ X of odd multiplicity.

First assume that 𝔦_{1}(X) = 1, so dim_{Izh} X = dim X. In this special case, we simultaneously prove both statements of Theorem 76.1 by induction on n = dim X + dim Y.

If n = 0, i.e., X and Y are both of dimension zero, then X = Spec K and Y = Spec L for some field extensions K and L of F with [K : F] = 2 and [L : F] even. Taking the push-forward to Spec(F) of the correspondence α, we have

[K : F] · mult(α) = [L : F] · mult(α^{t}).

Since mult(α) is odd, the correspondence α^{t} : Y ⇝ X is of odd multiplicity as needed.

So we may assume that n > 0. Let d be the dimension of X. We first prove (2), so we have dim Y = d > 0. It suffices to show that mult(α^{t}) is odd. Assume that the multiplicity of α^{t} is even. Let x ∈ X be a closed point of degree 2. Since the multiplicity of the correspondence [Y × x] : Y ⇝ X is 2 and the multiplicity of [x × Y] : X ⇝ Y is zero, modifying α by adding an appropriate multiple of [x × Y], we can assume that mult(α) is odd and mult(α^{t}) = 0.

The degree of the pull-back of α^{t} on X_{F (Y)} is now zero by Lemma 75.1. By Corollary 71.4, the degree homomorphism

deg : CH_{0}(X_{F (Y)}) → ℤ

is injective. Therefore, by Proposition 52.9, there is a nonempty open subset U ⊂ Y such that the restriction of α on X × U is trivial. Write Y′ for the reduced scheme Y \ U, and let i : X × Y′ → X × Y and j : X × U → X × Y denote the closed and open embeddings respectively. The sequence

CH_{d}(X × Y′) ⟶^{i_{∗}} CH_{d}(X × Y) ⟶^{j^{∗}} CH_{d}(X × U)

is exact by Proposition 57.9. Hence there exists an α′ ∈ CH_{d}(X × Y′) such that i_{∗}(α′) = α. We can view α′ as a correspondence X ⇝ Y′. Clearly, mult(α′) = mult(α), hence mult(α′) is odd. Since α′ is a linear combination of prime correspondences, there exists a prime correspondence β : X ⇝ Y′ of odd multiplicity. The class β is represented by a prime cycle, hence we may assume that Y′ is irreducible. Since dim Y′ < dim Y = dim X = dim_{Izh} X, we contradict statement (1) for the varieties X and Y′ that holds by the induction hypothesis.

We now prove (1) when 𝔦_{1}(X) = 1. Assume that dim Y < dim X. Let Z ⊂ X ×Y be a prime cycle representing α. Since mult(α) is odd, the projection Z → X is surjective and the field extension F (X) ↪ F (Z) is of odd degree. The restriction of the projection X × Y → Y defines a proper morphism Z → Y. Replacing Y by the image of this morphism, we assume that Z → Y is a surjection.

In view of Lemma 74.4, extending scalars to a purely transcendental extension of F, we can find a smooth subquadric X′ of X of the same dimension as Y having 𝔦_{1}(X′) = 1. By Lemma 75.2, all closed points on Y are still of even degree. Since purely transcendental extensions do not change Witt indices by Lemma 7.15, we still have 𝔦_{1}(X) = 1.

By Corollary 75.3, there exists a correspondence X′ ⇝ Y of odd multiplicity. Since dim X′ < dim X, by the induction hypothesis, statement (2) holds for X′ and Y, that is, X′ has a point over Y, i.e., there exists a rational morphism Y −−→ X′. Composing this morphism with the embedding of X′ into X, we get a rational morphism f : Y −−→ X.

Consider the rational morphism

h := id_{X} × f : X × Y −−→ X × X.

As the projection of Z to Y is surjective, Z intersects the domain of the definition of h. Let Z′ be the closure of the image of Z under h. The composition of Z −−→ Z′ with the first projection to X yields a tower of field extensions F (X) ⊂ F (Z′) ⊂ F (Z). As [F (Z) : F (X)] is odd, so is [F (Z′) : F (X)], i.e., the correspondence β : X ⇝ X given by Z′ is of odd multiplicity. The image of the second projection Z′ → X is contained in X′, hence mult(β^{t}) = 0 as dim X′ < dim X. This contradicts Theorem 75.4 and establishes Theorem 76.1 in the case 𝔦_{1}(X) = 1.

We now consider the general case. Let X′ be a smooth subquadric of X with dim X′ = dim_{Izh} X. Then 𝔦_{1}(X′) = 1 by Corollary 74.3, i.e., dim_{Izh} X′ = dim_{Izh} X. By Corollary 75.3, the scheme Y_{F (X′)} has a closed point of odd degree since Y_{F (X)} does. As 𝔦_{1}(X′) = 1, we have shown in the first part of the proof that the statements (1) and (2) hold for X′ and Y. In particular, dim_{Izh} X = dim_{Izh} X′ ≤ dim Y by (1) for X′ and Y proving (1) for X and Y. If dim X′ = dim Y, it follows from (2) applied to X′ and Y that X′ is isotropic over F (Y). Hence X is isotropic over F (Y) proving (2) for X and Y. □

Remark 76.2. The statement of Theorem 76.1 with X replaced by an anisotropic nonsmooth quadric (over a field of characteristic 2) was proved by Totaro in [132].

A consequence of Theorem 76.1 is that an anisotropic smooth quadric X cannot be compressed to a variety Y of dimension smaller than dim_{Izh} X with all closed points of even degree:

Corollary 76.3. Let X be an anisotropic smooth projective F -quadric and let Y be a complete F -variety with all closed points of even degree. If dim_{Izh} X > dim Y, then there are no rational morphisms X −−→ Y.

Remark 76.4. Let X and Y be as in part (2) of Theorem 76.1. Suppose in addition that dim X = dim_{Izh} X, i.e., 𝔦_{1}(X) = 1. Let α : X ⇝ Y be a correspondence of odd multiplicity. The proof of Theorem 76.1 shows mult(α^{t}) is also odd.

Applying Theorem 76.1 to the special (but perhaps the most interesting) case where the variety Y is also a projective quadric, we prove the conjectures of Izhboldin:

Theorem 76.5. Let X and Y be anisotropic smooth projective quadrics over F. Suppose that Y is isotropic over F (X). Then

(1) dim_{Izh} X ≤ dim_{Izh} Y.

(2) dim_{Izh} X = dim_{Izh} Y if and only if X is isotropic over F (Y).

Proof. Choose a subquadric Y′ ⊂ Y with dim Y′ = dim_{Izh} Y. Since Y′ becomes isotropic over F (Y) by Lemma 74.1(2) and Y becomes isotropic over F (X), the quadric Y′ becomes isotropic over F (X). By Theorem 76.1, we have dim_{Izh} X ≤ dim Y′. Moreover, in the case of equality, X becomes isotropic over F (Y′) and hence over F (Y). Conversely, if X is isotropic over F (Y), interchanging the roles of X and Y, the argument above also yields dim_{Izh} Y ≤ dim_{Izh} X, hence equality holds. □

We have the following upper bound for the Witt index of Y over F (X).

Corollary 76.6. Let X and Y be anisotropic smooth projective quadrics over F. Suppose that Y is isotropic over F (X). Then

𝔦_{0}(Y_{F (X)}) − 𝔦_{1}(Y) ≤ dim_{Izh} Y − dim_{Izh} X.

Proof. If dim_{Izh} X = 0, the statement is trivial. Otherwise, let Y′ be a smooth subquadric of Y of dimension dim_{Izh} X − 1. Since dim_{Izh} Y′ ≤ dim Y′ < dim_{Izh} X, the quadric Y′ remains anisotropic over F (X) by Theorem 76.5(1). Therefore, 𝔦_{0}(Y_{F (X)}) ≤ codim_{Y} Y′ = dim Y −dim_{Izh} X+1 by Lemma 74.1, hence the inequality holds. □

We also have the following more precise version of Theorem 76.1:

Corollary 76.7. Let X be an anisotropic smooth projective F -quadric and let Y be a complete variety over F such that every closed point of Y is of even degree. If there is a closed point in Y_{F (X)} of odd degree, then there exists a closed subvariety Y′ ⊂ Y such that

(1) dim Y′ = dim_{Izh} X.

(2) Y′_{F (X)} possesses a closed point of odd degree.

(3) X_{F (Y′)} is isotropic.

Proof. Let X′ ⊂ X be a smooth subquadric with dim X′ = dim_{Izh} X. Then dim_{Izh} X′ = dim X′ by Corollary 74.3. An odd degree closed point on Y_{F (X)} determines a correspondence X ⇝ Y of odd multiplicity which in turn gives a correspondence X′ ⇝ Y of odd multiplicity. We may assume that the latter correspondence is prime and take a prime cycle Z ⊂ X′ × Y representing it. Let Y′ be the image of the proper morphism Z → Y. Clearly, dim Y′ ≤ dim Z = dim X′ = dim_{Izh} X. On the other hand, Z determines a correspondence X′ ⇝ Y′ of odd multiplicity. Therefore dim Y′ ≥ dim X′ by Theorem 76.1, and condition (1) of Corollary 76.7 is satisfied. Moreover, Y′_{F (X′)} has a closed point of odd degree. Since the field F (X × X′) is a purely transcendental extension of F (X), Lemma 75.2 shows that there is a closed point on Y′_{F (X)} of odd degree, i.e., condition (2) of Corollary 76.7 is satisfied. Finally, the quadric X′_{F (Y′)} is isotropic by Theorem 76.1; therefore X_{F (Y′)} is isotropic. □

77. Addendum: The Pythagoras number

Given a field F, its pythagoras number is defined to be

p(F) := min {n | D(n⟨1⟩) = D(∞⟨1⟩)}

or infinity if no such integer exists. If char F = 2, then p(F) = 1 and if char F ≠ 2, then p(F) = 1 if and only if F is pythagorean. Let F be a field that is not formally real. Then the quadratic form (s(F)+1) ⟨1⟩ is isotropic. In particular, p(F) = s(F) or s(F) + 1 and each value is possible. So this invariant is only interesting when the field is formally real. For a given formally real field determining its pythagoras number is not easy. If F is an extension of a real closed field of transcendence degree n, then p(F) ≤ 2^{n} by Corollary 35.15. In particular, if n = 1 and F is not pythagorean then p(F) = 2. It is known that p(ℝ(t_{1}, t_{2})) = 4 (cf. [23]), but in general, the value of p(ℝ(t_{1}, …, t_{n})) is not known. In this section, given any nonnegative integer n, we construct a formally real field having pythagoras number n. This was first done by Hoffmann in [56].

Lemma 77.1. Let F be a formally real field and let φ be a quadratic form over F. If P ∈ 𝔛(F), then P extends to an ordering on F (φ) if and only if φ is indefinite at P, i.e., |sgn_{P} (φ)| < dim φ.

Proof. Suppose that φ is indefinite at P. Let F_{P} be the real closure of F with respect to P. Let K = F_{P} (φ). As φ_{F_{P}} is isotropic, K/F_{P} is purely transcendental. Therefore the unique ordering on F_{P} extends to K. The restriction of this extension to F (φ) extends P. The converse is clear. □

The following proposition is a consequence of the lemma and Theorem 76.5.

Proposition 77.2. Let F be formally real and let x, y ∈ D(∞⟨1⟩). Let φ ≃ m⟨1⟩ ⊥ ⟨−x⟩ and ψ ≃ n⟨1⟩ ⊥ ⟨−y⟩ with n > m ≥ 0. Then F (ψ) is formally real. If, in addition, φ is anisotropic, then so is φ_{F (ψ)}.

Proof. As ψ is indefinite at every ordering, every ordering of F extends to F (ψ). In particular, F (ψ) is formally real. Suppose that φ is anisotropic. Since over each real closure of F both φ and ψ have Witt index 1, the first Witt index of φ and ψ must also be 1. As dim φ > dim ψ, the form φ_{F (ψ)} is anisotropic by Theorem 76.5. □

Let F_{0} be a formally real field. Let F_{1} = F_{0}(t_{1}, …, t_{n−1}) and let x = 1 + t_{1}^{2} + ⋯ + t_{n−1}^{2} ∈ D(∞⟨1⟩). By Corollary 17.13, the element x is a sum of n squares in F_{1} but no fewer. In particular, φ ≃ (n − 1)⟨1⟩ ⊥ ⟨−x⟩ is anisotropic over F_{1}. For i ≥ 1, inductively define F_{i+1} as follows:

Let

𝔄_{i} := {n⟨1⟩ ⊥ ⟨−y⟩ | y ∈ D(∞⟨1⟩_{F_{i}}) }.

For any finite subset S ⊂ 𝔄_{i}, let X_{S} be the product of quadrics X_{φ} for all φ ∈ S. If S ⊂ T are two subsets of 𝔄_{i}, we have the dominant projection X_{T} → X_{S} and therefore the inclusion of function fields F (X_{S}) → F (X_{T}). Set F_{i+1} = colim F_{S} over all finite subsets S ⊂ 𝔄_{i}. By construction, all quadratic forms φ ∈ 𝔄_{i} are isotropic over the field extension F_{i} of F. Let F = ⋃ F_{i}. Then F has the following properties:

(1) F is formally real.

(2) n⟨1⟩ ⊥ ⟨−y⟩ is isotropic for all 0 ≠ y ∈ ∑(F^{×})^{2}. Consequently, D(∞⟨1⟩_{F} ) ⊂ D(n⟨1⟩_{F} ), so the pythagoras number p(F) ≤ n. As φ ≃ (n − 1) ⟨1⟩ ⊥ ⟨−x⟩ remains anisotropic over F, we have p(F) ≥ n. So we have proved:

Theorem 77.3. For every n ≥ 1 there exists a formally real field F with p(F) = n.

CHAPTER XV Application of Steenrod Operations

Since Steenrod operations are not available in characteristic 2, throughout this chapter, the characteristic of the base field is assumed to be different from 2.

We write v_{2}(n) for the 2-adic exponent of an integer n.

We shall use the notation of Chapter XIII. In particular, X is a smooth Ddimensional projective quadric over a field F given by a (nondegenerate) quadratic form φ, and d = [D/2].

78. Computation of Steenrod operations

Recall that h ∈ Ch^{1}(X) is the class of a hyperplane section.

Lemma 78.1. The modulo 2 total Chern class c(T_{X}) : Ch(X) → Ch(X) of the tangent vector bundle T_{X} of the quadric X is multiplication by (1 + h)^{D+2}.

Proof. By Proposition 58.15, it suffices to show that c(T_{X})([X]) = (1+h)^{D+2}. Let i : X ↪ ℙ be the closed embedding of X into the (D+1)-dimensional projective space ℙ = ℙ(V), where V is the underlying vector space of φ. We write H ∈ Ch^{1}(ℙ) for the class of a hyperplane, so h = i^{∗}(H). Since X is a hypersurface in ℙ of degree 2, the normal bundle N of the embedding i is isomorphic to i^{∗}(L^{⊗2}), where L is the canonical line bundle over ℙ (cf. §104.B). By Propositions 104.16 and 54.7, we have c(T_{X}) ∘ c(i^{∗}(L^{⊗2})) = c(i^{∗}T_{ℙ}). By Example 61.16, we know that c(T_{ℙ}) is the multiplication by (1+H)^{D+2} and by Propositions 54.3 and 57.26, that c(L^{⊗2}) = id modulo 2. It follows that

c(T_{X})([X]) = (c(i^{∗}T_{ℙ}) ∘ c(i^{∗}(L^{⊗2}))^{−1})(i^{∗}([ℙ]))
= (i^{∗} ∘ c(T_{ℙ}) ∘ c(L^{⊗2})^{−1})([ℙ]) = i^{∗}(1 + H)^{D+2} = (1 + h)^{D+2}

by Proposition 55.22. □

Corollary 78.2. Suppose that 𝔦_{0}(X) > n for some n ≥ 0. Let W ⊂ V be a totally isotropic (n + 1)-dimensional subspace of V and let ℙ be the n-dimensional projective space ℙ(W). Let i : ℙ ↪ X be the closed embedding. Then the modulo 2 total Chern class c(N) : Ch(ℙ) → Ch(ℙ) of the normal bundle N of the embedding i is multiplication by (1 + H)^{D+1−n}, where H ∈ Ch^{1}(ℙ) is the class of a hyperplane.

Proof. By Propositions 104.16 and 54.7, we have c(N) = c(T_{ℙ})^{−1} ∘ c(i^{∗}T_{X}); and by Proposition 55.22 and Lemma 78.1, we have

c(i^{∗}T_{X}[ℙ]) = c(i^{∗}T_{X})(i^{∗}[X]) = (i^{∗} ∘ c(T_{X}))([X]) = i^{∗}(1 + h)^{D+2} = (1 + H)^{D+2}.

Hence by Example 61.16, c(T_{ℙ}) = (1 + H)^{n+1}. □

Corollary 78.3. Under the hypothesis of Corollary 78.2, we have

Sq_{X}([ℙ]) = [ℙ] · (1 + h)^{D+1−n}.

Proof. By the Wu Formula (Proposition 61.8), Sq_{X}([ℙ]) = i_{∗}(c(N)[ℙ]). By Corollary 78.2, we get

i_{∗}(c(N)[ℙ]) = i_{∗}((1+H)^{D+1−n}·[ℙ]) = i_{∗}(i^{∗}(1+h)^{D+1−n}·[ℙ]) = (1+h)^{D+1−n}·i_{∗}[ℙ]

using the Projection Formula (Proposition 56.9). □

We also have (cf. Example 61.16):

Lemma 78.4. Sq_{X}(h^{i}) = h^{i} · (1 + h)^{i} for any i ≥ 0.

Corollary 78.5. If the quadric X is split, then the ring endomorphism Sq_{X} : Ch(X) → Ch(X) acts on the basis {h^{i}, l_{i}}_{i∈[0, d]} of Ch(X) by the formulas

Sq_{X}(h^{i}) = h^{i} · (1 + h)^{i} and Sq_{X}(l_{i}) = l_{i} · (1 + h)^{D+1−i}.

In particular, for any j ≥ 0,

Sq_{X}^{j}(h^{i}) = \binom{i}{j}h^{i+j} and Sq_{X}^{j}(l_{i}) = \binom{D + 1 − i}{j}l_{i−j}.

Binomial coefficients modulo 2 are computed as follows (we leave the proof to the reader): Let ℕ be the set of nonnegative integers, let 2^{ℕ} be the set of all subsets of ℕ, and let π : ℕ → 2^{ℕ} be the injection given by base 2 expansions. For any n ∈ ℕ, the set π(n) consists of all those m ∈ ℕ such that the base 2 expansion of n has 1 in the mth position. For two arbitrary nonnegative integers i and n, write i ⊂ n if π(i) ⊂ π(n).

Lemma 78.6. For any i, n ∈ ℕ, the binomial coefficient \binom{n}{i} is odd if and only if i ⊂ n.

79. Values of the first Witt index

The main result of this section is Theorem 79.9 (conjectured by Hoffmann and originally proved in [72]); its main ingredient is given by Proposition 79.4. We begin with some observations.

Remark 79.1. By Theorem 61.9,

Ch(X^{∗}) ⟶^{Sq_{X^{∗}}} Ch(X^{∗})
↓  ↓
Ch(X̄^{∗}) ⟶^{SqX̄^{∗}} Ch(X̄^{∗})

is commutative, hence we get an endomorphism \overline{Ch}(X^{∗}) → \overline{Ch}(X^{∗}) that we shall also call a Steenrod operation and denote it by Sq_{X^{∗}}, even though it is a restriction of SqX̄^{∗} and not of Sq_{X^{∗}}.

Remark 79.2. Let l_{n} × h^{m} ∈ Ch(X̄^{2}) be an essential basis element with n ≥ m. Since Sq(l_{n} × h^{m}) = Sq(l_{n}) × Sq(h^{m}) by Theorem 61.14, we see by Corollary 78.5, that the value of Sq(l_{n} × h^{m}) is a linear combination of the elements l_{i} × h^{j} with i ≤ n and j ≥ m. If m = 0, one can say more: Sq(l_{n} × h^{0}) is a linear combination of the elements l_{i} × h^{0} with i ≤ n.

Of course, we have similar facts for the essential basis elements of type h^{m} ×l_{n}. Representing essential basis elements of type l_{n}×h^{m} with n ≥ m as points of the

right pyramid of Remark 73.8, we may interpret the above statements graphically as follows: the diagram of the value of the Steenrod operation on a point l_{n} × h^{m} is contained in the isosceles triangle based on the lower row of the pyramid whose top is the point l_{n} × h^{m} (an example of this is the picture on the left below). If l_{n} × h^{m} is on the right side of the pyramid, then the diagram of the value of the Steenrod operation is contained in the part of the right side of the pyramid, which is below the point (an example of this is the picture on the right below).

∘  ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ • ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ • ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘  ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗

The next statement follows simply from Remark 79.2.

Lemma 79.3. Assume that X is anisotropic. Let π ∈ Ch_{D+𝔦_{1}−1}(X^{2}) be the 1primordial cycle. For any j ≥ 1, the element S_{X^{2}}^{j}(π) has no points in the first shell triangle.

Proof. By the definition of π, the only point the cycle π has in the first (left as well as right) shell triangle is the top of the triangle. By Remark 79.2, the only point in the first left shell triangle that may be contained in S^{j}(π) is the point on the left side of the triangle and the only point in the first right shell triangle that may be contained in S^{j}(π) is the point on the right side of the triangle. Since these two points are not dual (points on the left side of the first left shell triangle are dual to points on the left side of the first right shell triangle), the lemma follows by Corollary 73.21. □

We shall obtain further information in Lemma 83.1 below. We write exp_{2}(a) for 2^{a}.

Proposition 79.4. For any anisotropic quadratic form φ of dim φ ≥ 2,

𝔦_{1}(φ) ≤ exp_{2} v_{2}( dim φ − 𝔦_{1}(φ)).

Proof. Let r = v_{2}(dim φ − 𝔦_{1}(φ)). Apply the Steenrod operation Sq_{X^{2}}^{2^{r}} : \overline{Ch}(X^{2}) → \overline{Ch}(X^{2}) to the 1-primordial cycle π. Since

Sq_{X^{2}}^{2^{r}}(h^{0} × l_{𝔦_{1}−1}) = h^{0} × Sq_{X}^{2^{r}}(l_{𝔦_{1}−1}) = \binom{dim φ − 𝔦_{1}}{2^{r}} · (h^{0} × l_{𝔦_{1}−1−2^{r}})

by Theorem 61.14 and Corollary 78.5 and the binomial coefficient is odd by Lemma 78.6, we have h^{0} × l_{𝔦_{1}−1−2^{r}} ∈ Sq_{X^{2}}^{2^{r}}(α). It follows by Lemma 79.3 that 2^{r} ∉ [1, 𝔦_{1}), i.e., that 2^{r} ≥ 𝔦_{1}. □

Remark 79.5. Let a be a positive integer written in base 2. A suffix of a is an integer written in base 2 that is obtained from a by deleting several (and at least one) consecutive digits starting from the left one. For example, all suffixes of 1011010 are 11010, 1010, 10 and 0.

Let i < n be two nonnegative integers. Then the following are equivalent.

(1) i ≤ exp_{2} v_{2}(n − i).

(2) There exists an r ≥ 0 satisfying 2^{r} < n, i ≡ n (mod 2^{r}), and i ∈ [1, 2^{r}].

(3) i − 1 is the remainder upon division of n − 1 by an appropriate 2-power.

(4) The 2-adic expansion of i − 1 is a suffix of the 2-adic expansion of n − 1.

(5) The 2-adic expansion of i is a suffix of the 2-adic expansion of n or i is a 2-power divisor of n.

In particular, the integers i = 𝔦_{1}(φ) and n = dim φ in Proposition 79.4 satisfy these conditions.

Corollary 79.6. All relative higher Witt indices of an odd-dimensional quadratic form are odd. Any relative higher Witt index of an even-dimensional quadratic form is either even or 1.

Example 79.7. Assume that φ is anisotropic and let s ≥ 0 be the largest integer satisfying dim φ > 2^{s}. Then it follows by Proposition 79.4 that 𝔦_{1}(φ) ≤ dim φ − 2^{s} (e.g., by Condition (4) of Remark 79.5). In particular, if dim φ = 2^{s} + 1, then 𝔦_{1}(φ) = 1.

The first statement of the following corollary is the Separation Theorem 26.5 (over a field of characteristic not 2); the second statement was originally proved by O. Izhboldin (using a different method) in [63, Th. 02]. A characteristic 2 version has been proven by Hoffmann and Laghribi in [58, Th. 1.3].

Corollary 79.8. Let φ and ψ be two anisotropic quadratic forms over F. (1) If dim ψ ≤ 2^{s} < dim φ for some s ≥ 0, then the form ψ_{F (φ)} is anisotropic.

(2) Suppose that dim ψ = 2^{s} + 1 ≤ dim φ for some s ≥ 0. If the form ψ_{F (φ)} is isotropic, then the form φ_{F (ψ)} is also isotropic.

Proof. Let X and Y be the quadrics of φ and of ψ respectively. Then dim_{Izh} X ≥ 2^{s} −1 by Example 79.7. If dim ψ ≤ 2^{s}, then dim Y ≤ 2^{s} −2. Therefore,

dim_{Izh} Y ≤ dim Y < 2^{s} − 1 ≤ dim_{Izh} X

and Y_{F (X)} is anisotropic by Theorem 76.5(1).

Suppose that dim ψ = 2^{s} + 1. Then dim_{Izh} Y = 2^{s} − 1 ≤ dim_{Izh} X. If Y_{F (X)} is isotropic, then dim_{Izh} Y = dim_{Izh} X by Theorem 76.5(1) and therefore X_{F (Y)} is isotropic by Theorem 76.5(2). □

We show next that all values of the first Witt index not forbidden by Proposition 79.4 are possible by establishing the main result of this section:

Theorem 79.9. Two nonnegative integers i and n satisfy i ≤ exp_{2} v_{2}(n − i) if and only if there exists an anisotropic quadratic form φ over a field of characteristic not 2 with

dim φ = n and 𝔦_{1}(φ) = i.

Proof. By Proposition 79.4 and Remark 79.5, we need only prove the necessity. Let i and n be two nonnegative integers satisfying i ≤ exp_{2} v_{2}(n − i). Let r be as in condition (2) of Remark 79.5. Write n − i = 2^{r} · m for some integer m.

Let F be any field of characteristic not 2 and consider the field

K = F (t_{1}, …, t_{r})

of rational functions in r variables. By Corollary 19.6, the Pfister form π = ⟨⟨t_{1}, …, t_{r}⟩⟩ is anisotropic over K. Let L = K(s_{1}, …, s_{m}), where s_{1}, …, s_{m} are variables. By Lemma 19.5, the quadratic L-form ψ = π_{L} ⊗ ⟨1, s_{1}, …, s_{m}⟩ is anisotropic.

We claim that 𝔦_{1}(ψ) = 2^{r}. Indeed, by Proposition 6.22, we have 𝔦_{1}(ψ) ≥ 2^{r}. On the other hand, the field E = L(\sqrt{−s_{1}}) is purely transcendental over K(s_{2}, …, s_{m}) and therefore 𝔦_{0}(ψ_{E}) = 2^{r}. Consequently, 𝔦_{1}(ψ) = 2^{r}.

Let φ be an arbitrary subform of ψ of codimension 2^{r} − i. Since dim ψ = 2^{r} · (m + 1) = n + (2^{r} − i), the dimension of φ is equal to n. As 2^{r} − i < 2^{r} = 𝔦_{1}(ψ), we have 𝔦_{1}(φ) = i by Corollary 74.3. □

80. Rost correspondences

Recall that by abuse of notation we also denote the image of the element h ∈ CH^{1}(X) in the groups CH^{1}(X̄), Ch^{1}(X), and Ch^{1}(X̄) by the same symbol h. In the following lemma, h stands for the element of Ch(X).

Lemma 80.1. Let n be the integer satisfying

2^{n} − 1 ≤ D ≤ 2^{n+1} − 2.

Set s = D − 2^{n} + 1 and r = 2^{n+1} − 2 − D (observe that r + s = 2^{n} − 1). If α ∈ Ch_{r+s}(X), then

Sq_{r+s}^{X}(α) = h^{r} · α^{2} ∈ Ch_{0}(X).

Proof. By the definition of the cohomological Steenrod operation Sq_{X}, we have Sq_{X} = c(T_{X})∘Sq^{X}, where Sq^{X} is the homological Steenrod operation. Therefore, Sq^{X} = c(−T_{X}) ∘ Sq_{X}. In particular,

Sq_{r+s}^{X}(α) = ∑_{i=0}^{r+s} c_{i}(−T_{X}) ∘ Sq_{X}^{r+s−i}(α)

in Ch_{0}(X). By Lemma 78.1, we have c_{i}(−T_{X}) = \binom{−D−2}{i} · h^{i}. As \binom{−D−2}{i} = ±\binom{D+i+1}{i}, it follows from Lemma 78.6, that the latter binomial coefficient is even for any i ∈ [r + 1, r + s] and is odd for i = r. Since Sq_{X}^{r+s−i}(α) is equal to 0 for i ∈ [0, r − 1] and is equal to α^{2} for i = r by Theorem 61.13, the required relation is established. □

Theorem 80.2. Let n be the integer satisfying

2^{n} − 1 ≤ D ≤ 2^{n+1} − 2.

Set s = D − 2^{n} + 1 and r = 2^{n+1} − 2 − D. Let X and Y be two anisotropic projective quadrics of dimension D over a field of characteristic not 2. Let ρ̄ be a cycle in \overline{Ch}_{r+s}(X × Y). Then (pr_{X})_{∗}(ρ̄) = 0 if and only if (pr_{Y})_{∗}(ρ̄) = 0, where pr_{X} : X × Y → X and pr_{Y} : X × Y → Y are the projections.

Proof. Let ρ be an element of the nonreduced Chow group Ch_{r+s}(X × Y). Write ρ̄ for the image of ρ in \overline{Ch}_{r+s}(X × Y). The group \overline{Ch}_{r+s}(X) is generated by h^{s} = h_{X}^{s} (if s = d, this is true as X is anisotropic, hence not split). Therefore, we have (pr_{X})_{∗}(ρ̄) = a_{X}h_{X}^{s} for some a_{X} ∈ ℤ/2ℤ. Similarly, (pr_{Y})_{∗}(ρ̄) = a_{Y} h_{Y}^{s} for some a_{Y} ∈ ℤ/2ℤ. To prove Theorem 80.2 we must show a_{X} = a_{Y}.

As X and Y are anisotropic, any closed point z in X or Y has even degree. In particular, the map

\frac{1}{2} deg_{Z} : Ch_{0}(Z) → ℤ/2ℤ

given by z ↦ \frac{1}{2} deg(z) (mod 2) is well-defined for Z = X or Y. Consider the following diagram:

(80.3)  [figure: Ch_{r+s}(X × Y); Ch_{r+s}(X); Ch_{r+s}(Y); Ch_{0}(X × Y); Ch_{0}(X); Ch_{0}(Y); ℤ/2ℤ; (pr_{X})_{∗}; (pr_{Y} )_{∗}; Sq_{r+s}^{X×Y}; Sq_{r+s}^{X}; Sq_{r+s}^{Y}; (pr_{X})_{∗}; (pr_{Y} )_{∗}; \frac{1}{2} deg_{X}; \frac{1}{2} deg_{Y}]

We show that the diagram (80.3) is commutative. The bottom diamond is commutative by the functorial property of the push-forward homomorphism (cf. Proposition 49.9 and Example 57.7). The left and the right parallelograms are commutative by Theorem 60.5. Therefore,

(\frac{1}{2} deg_{X}) ∘ Sq_{r+s}^{X} ∘(pr_{X})_{∗}(ρ) = (\frac{1}{2} deg_{Y}) ∘ Sq_{r+s}^{Y} ∘(pr_{Y})_{∗}(ρ).

Applying Lemma 80.1 to the element α = (pr_{X})_{∗}(ρ), we have

(\frac{1}{2} deg_{X}) ∘ Sq_{r+s}^{X} ∘(pr_{X})_{∗}(ρ) = (\frac{1}{2} deg_{X})(h_{X}^{r} · α^{2}) = a_{X}.

Similarly (\frac{1}{2} deg_{Y}) ∘ Sq_{r+s}^{Y} ∘(pr_{Y})_{∗}(ρ) = a_{Y}, proving the theorem. □

Exercise 80.4. Use Theorem 80.2 to prove the following generalization of Corollary 79.8(2): Let X and Y be two anisotropic projective quadrics satisfying dim X = dim Y = D. Let s be as in Theorem 80.2. If there exists a rational morphism X −−→ Y, then there exists a rational morphism G_{s}(Y) −−→ G_{s}(X) where for an integer i, G_{i}(X) is the scheme (variety, if i ≠ D/2) of i-dimensional linear subspaces lying on X. (We shall study the scheme G_{d}(X) in Chapter XVI.)

Remark 80.5. One can generalize Theorem 80.2 as follows. We replace Y by an arbitrary projective variety of an arbitrary dimension (and, in fact, Y need not be smooth nor of dimension D = dim X). Suppose that every closed point of Y has even degree. Let ρ ∈ Ch_{r+s}(X × Y) satisfy (pr_{X})_{∗}(ρ̄) ≠ 0 ∈ \overline{Ch}(X). Then (pr_{Y})_{∗}(ρ) ≠ 0 ∈ Ch(Y) (note that this is in Ch(Y), not \overline{Ch}(Y)). To prove this generalization, we use the commutative diagram 80.3. As before, we have deg_{X} ∘ Sq_{r+s}^{X} ∘(pr_{X})_{∗}(ρ) ≠ 0 provided that pr_{X}(ρ̄) ≠ 0. Therefore,

deg_{Y} ∘ Sq_{r+s}^{Y} ∘(pr_{Y})_{∗}(ρ) ≠ 0.

In particular, (pr_{Y})_{∗}(ρ) ≠ 0.

Exercise 80.6. Show that one cannot replace the conclusion (pr_{Y})_{∗}(ρ) ≠ 0 ∈ Ch(Y) by (pr_{Y})_{∗}(ρ̄) ≠ 0 ∈ \overline{Ch}(Y) in Remark 80.5. (Hint: Let Y be an anisotropic quadric with X a subquadric of Y satisfying 2 dim X < dim Y and ρ ∈ Ch(X × Y) the class of the diagonal of X.)

Taking Y = X in Theorem 80.2, we have

Corollary 80.7. Let X be an anisotropic quadric of dimension D and let s be as in Theorem 80.2. If a rational cycle in Ch(X̄^{2}) contains h^{s} × l_{0}, then it also contains l_{0} × h^{s}.

Corollary 80.8. Assume that X is an anisotropic quadric of dimension D and for some integer i ∈ [0, d] the cycle h^{0} × l_{i} + l_{i} × h^{0} ∈ Ch(X̄^{2}) is rational. Then the integer dim X − i + 1 is a power of 2.

Proof. If the cycle h^{0} × l_{i} + l_{i} × h^{0} is rational, then, multiplying by h^{s} × h^{i}, we see that the cycle h^{s} ×l_{0} +l_{i−s} ×h^{i} is also rational. By Corollary 80.7, it follows that i = s. Therefore, dim X − i + 1 = 2^{n} with n as in Theorem 80.2. □

Remark 80.9. By Lemma 73.12 and Corollary 73.21, the integer i in Corollary 80.8 is necessarily equal to 𝔦_{1}(X) − 1.

Recalling Definition 73.29, we have

Corollary 80.10. If dim φ − 𝔦_{1}(φ) is not a 2-power, then the 1-primordial cycle on X^{2} produces an integer.

Proof. If the 1-primordial cycle π does not produce any integer, then π = h^{0}×l_{𝔦_{1}−1}+l_{𝔦_{1}−1}×h^{0}. Therefore, by Corollary 80.8, the integer D−(𝔦_{1}(φ)−1)+1 = dim φ − 𝔦_{1}(φ) is a 2-power. □

The element h^{0} × l_{0} + l_{0} × h^{0} ∈ Ch(X̄^{2}) is called the Rost correspondence of the quadric X.

Of course, the Rost correspondence of an isotropic quadric X is rational. A special case of Corollary 80.8 is given by:

Corollary 80.11. If X is anisotropic and the Rost correspondence of X is rational, then D + 1 is a power of 2.

By multiplying by h^{1} × h^{0}, we see that rationality of the Rost correspondence implies rationality of the element h^{1} ×l_{0}. In fact, rationality of h^{1} ×l_{0} alone implies that D + 1 is a power of 2:

Corollary 80.12. If X is anisotropic and the element h^{1}×l_{0} ∈ Ch(X̄^{2}) is rational, then D + 1 is a power of 2.

Proof. If h^{1} × l_{0} is rational, then for any i ≥ 1, the element h^{i} × l_{0} is also rational. Let s be as in Theorem 80.2. By Corollary 80.7 it follows that s = 0, i.e., D = 2^{n} − 1. □

Let A be a point of a shell triangle of a quadric. We write A^{♯} for the dual point in the sense of Definition 73.20. The following statement was originally proved (in characteristic 0) by Vishik.

Corollary 80.13. Let Y be another anisotropic projective quadric of dimension D. Basis elements of Ch(Ȳ^{2}) are in natural 1-1 correspondence with basis elements of Ch(X̄^{2}). Assume that 𝔦_{1}(Y) = s+1 with s as in Corollary 80.7. Let A be a point of a first shell triangle of Y and let A^{♯} be its dual point. Then in the diagram of any element of \overline{Ch}(X^{2}), the point corresponding to A is marked if and only if the point corresponding to A^{♯} is marked.

Proof. We may assume that A lies in the left first shell triangle of Y. Let h^{i} × l_{j} (with 0 ≤ i ≤ j ≤ s) be the basis element represented by A. Then the basis element represented by A^{♯} is l_{s−i} × h^{s−j}. Let α ∈ \overline{Ch}(X^{2}) and assume that α contains h^{i} × l_{j}. Then the rational cycle (h^{s−i} × h^{j}) · α contains h^{s} × l_{0}. Therefore, by Corollary 80.7, this rational cycle also contains l_{0} ×h^{s}. It follows that α contains l_{s−i} × h^{s−j}. □

Remark 80.14. The equality 𝔦_{1}(Y) = s + 1 holds if Y is excellent. By Theorem 79.9 this value of the first Witt index is maximal for all D-dimensional anisotropic quadrics.

81. On the 2-adic order of higher Witt indices, I

The main result of this section is Theorem 81.2 on a relationship between higher Witt indices and the integer produced by a 1-primordial cycle originally proved in [75]. This is used to establish a relationship between higher Witt indices of an anisotropic quadratic form (cf. Corollary 81.19).

Let φ be a nondegenerate (possibly isotropic) quadratic form of dimension D over a field F of characteristic not 2 and X = X_{φ}. Let 𝔥 = 𝔥(φ) be the height of φ (or X) and let

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{𝔥}

be the generic splitting tower (cf. §25). For q ∈ [0, 𝔥], let 𝔦_{q} = 𝔦_{q}(φ), 𝔧_{q} = 𝔧_{q}(φ), φ_{q} = (φ_{F_{q}})_{an}, and X_{q} = X_{φ_{q}}.

We shall use the following simple observation in the proof of Theorem 81.2:

Proposition 81.1. Let α be a homogeneous element of Ch(X̄^{2}) with codim α > d. Assume X is not split, i.e., 𝔥 > 0 and that for some q ∈ [0, 𝔥 − 1] the cycle α is F_{q}-rational and contains neither h^{i} × l_{?} nor l_{?} × h^{i} for any i < 𝔧_{q}. Then δ_{X}^{∗}(α) = 0 in Ch(X̄), where δ_{X} : X → X^{2} is the diagonal morphism of X.

Proof. We may assume that dim α = D+i with i ≥ 0 (otherwise dim δ_{X}^{∗}(α) < 0). As X is not hyperbolic, l_{d} × l_{d} ∉ α by Lemma 73.2. Therefore, by Proposition 68.1, δ_{X}^{∗}(α) = nl_{i}, where n is the number of essential basis elements contained in α. Since α contains neither h^{i} × l_{?} nor l_{?} × h^{i} with i < 𝔧_{q}, the number of essential basis elements contained in α coincides with the number of essential basis elements contained in pr_{∗}^{2}(α), where

pr_{∗}^{2} : \overline{Ch}(X_{F_{q}}^{2}) → \overline{Ch}(X_{q}^{2})

is the homomorphism of Remark 72.5. In view of Corollary 72.4, the latter number is even by Lemma 73.14. □

We have defined minimal and primordial elements in \overline{Ch}(X^{2}) for an anisotropic quadric X (cf. Definitions 73.5 and 73.16). We extend these definitions to the case of an arbitrary quadric.

Let X be an arbitrary (smooth) quadric given by a quadratic form φ (not necessarily anisotropic) and let X_{0} be the quadric given by the anisotropic part of φ. The images of minimal (respectively primordial) elements via the embedding in_{∗}^{2} : \overline{Ch}(X_{0}^{2}) → \overline{Ch}(X^{2}) of Remark 72.5 are called minimal (respectively primordial) elements of \overline{Ch}(X^{2}).

Theorem 81.2 (cf. [75, Th. 5.1]). Let X be an anisotropic quadric of even dimension over a field of characteristic not 2. Let π ∈ \overline{Ch}(X^{2}) be the 1-primordial cycle. Suppose that π produces an integer q ∈ [2, 𝔥] satisfying v_{2}(𝔦_{2}+⋯+𝔦_{q−1}) ≥ v_{2}(𝔦_{1})+2. Then v_{2}(𝔦_{q}) ≤ v_{2}(𝔦_{1}) + 1.

Proof. We fix the following notation:

a = 𝔦_{1},
b = 𝔦_{2} + ⋯ + 𝔦_{q−1} = 𝔧_{q−1} − a,
c = 𝔦_{q}.

Set n = v_{2}(𝔦_{1}). Then v_{2}(b) ≥ n + 2.

Consider the (a − 1)th derivative α = π · (h^{0} × h^{a−1}) of π. By Lemma 73.10 and Proposition 73.27, the cycle α is minimal since π is minimal and α contains the basis elements h^{0} × l_{0} and h^{a+b} × l_{a+b}.

Suppose the result is false, i.e., v_{2}(c) ≥ n+2. Proposition 81.3 below contradicts the minimality of α, hence proving Theorem 81.2. To state Proposition 81.3, we need the following morphisms:

g_{1} : X_{F (X)}^{2} → X^{3},

the morphism given by the generic point of the first factor of X^{3};

t_{12} : \overline{Ch}(X^{3}) → \overline{Ch}(X^{3}),

the automorphism given by the transposition of the first two factors of X^{3};

δ_{X^{2}} : X^{2} → X^{4}, (x_{1}, x_{2}) ↦ (x_{1}, x_{2}, x_{1}, x_{2}),

the diagonal morphism of X^{2}. We also use the pairing

∘ : Ch(X̄^{r}) × Ch(X̄^{s}) → Ch(X̄^{r+s−2})

(for various r, s ≥ 1) given by a composition of correspondences, where the elements of Ch(X̄^{s}) are considered as correspondences X̄^{s−1} ⇝ X̄ and the elements of Ch(X̄^{r}) are considered as correspondences X̄ ⇝ X̄^{r−1}.

Note that applying Proposition 73.23 to the quadric X_{1} with cycle pr_{∗}^{2}(π) ∈ \overline{Ch}(X_{1}^{2}), there exists a homogeneous essential symmetric cycle β ∈ \overline{Ch}(X_{F (X)}^{2}) containing the basis element h^{a+b} × l_{a+b+c−1} and none of the basis elements having h^{i} with i < a + b as a factor.

Proposition 81.3. In the notation above let η ∈ \overline{Ch}(X^{3}) be a preimage of β under the pull-back epimorphism g_{1}^{∗} and let μ be the essence of the composition η∘α. Then the cycle

(h^{0} × h^{c−a−1}) · δ_{X^{2}}^{∗}(t_{12}(μ) ∘ ( Sq_{X^{3}}^{2a}(μ) · (h^{0} × h^{0} × h^{c−a−1}))) ∈ \overline{Ch}(X^{2})

contains h^{a+b} × l_{a+b} and does not contain h^{0} × l_{0}.

Proof. Recall that b ≥ 0 and 2^{n+2} divides b and c, where n = v_{2}(a). By Proposition 79.4, we also have that 2^{n+2} divides dim φ_{q−1}, so 2^{n+2} divides dim φ_{1}. We then see, again using Proposition 79.4, that a = 2^{n}. In addition, dim φ ≡ 2a (mod 2^{n+2}) so

(81.4)  Sq_{X}^{2a}(l_{a+b+c−1}) = 0

by Corollary 78.5 and Lemma 78.6.

Let Sym(ρ) = ρ + ρ^{t} for a cycle ρ on X̄^{2} be the symmetrization operation. The cycle β is homogeneous, essential, symmetric, and does not contain any basis element having h^{i} with i < a+b as a factor. Consequently, we can write β = β_{0}+β_{1} with

(81.5)  β_{0} = Sym (h^{a+b} × l_{a+b+c−1}),
(81.6)  β_{1} = Sym ( ∑_{i∈I} h^{i+a+b} × l_{i+a+b+c−1})

for some set of positive integers I. Furthermore, since α does not contain any of h^{i} × l_{i} with i ∈ (0, a + b), we have

(81.7)  μ = h^{0} × β + h^{a+b} × γ + ν

for some essential cycle γ ∈ Ch_{D+a+b+c−1}(X̄^{2}) and some cycle ν ∈ Ch(X̄^{3}) such that the first factor of every basis element included in ν is of codimension > a + b. We can decompose γ = γ_{0} + γ_{1} with

(81.8)  γ_{0} = x · (h^{0} × l_{a+b+c−1}) + y · (l_{a+b+c−1} × h^{0}),
(81.9)  γ_{1} = ∑_{j∈J} h^{j} × l_{j+a+b+c−1} + ∑_{j∈J′} l_{j+a+b+c−1} × h^{j}

for some modulo 2 integers x, y ∈ ℤ/2ℤ and some sets of integers J, J′ ⊂ (0, +∞). We need the following:

Lemma 81.10. In the above, we have x = y = 1, I ⊂ [c, +∞), and J, J′ ⊂ [a + b + c, +∞).

Proof. To determine y, consider the cycle δ^{∗}(μ)·(h^{0} ×h^{c−1}) ∈ \overline{Ch}(X^{2}) where δ : X^{2} → X^{3} is the morphism (x_{1}, x_{2}) ↦ (x_{1}, x_{2}, x_{1}). This rational cycle does not contain h^{0} × l_{0}, while the coefficient of h^{a+b} × l_{a+b} equals 1 + y. Consequently, y = 1 by the minimality of α.

Similarly, using the morphism X^{2} → X^{3}, (x_{1}, x_{2}) ↦ (x_{1}, x_{1}, x_{2}) instead of δ, one sees that x = 1 (although the value of x is not important for our future purposes).

To show that I ⊂ [c, +∞), assume to the contrary that 0 < i < c for some i ∈ I. Then l_{i+a+b} ∈ \overline{Ch}(X_{F_{q}}) for this i and therefore the cycle

l_{i+a+b+c−1} = (pr_{3})_{∗}((l_{0} × l_{i+a+b} × h^{0}) · μ)

is F_{q}-rational, where pr_{3} : X^{3} → X is the projection onto the third factor. This contradicts Corollary 72.6 because i + a + b + c − 1 ≥ a + b + c = 𝔧_{q}(X) = 𝔦_{0}(X_{F_{q}}).

To prove the statement for J, assume to the contrary that there exists a j ∈ J with 0 < j < a + b + c. Then l_{j} ∈ \overline{Ch}(X_{F_{q}}), hence

l_{j+a+b+c−1} = (pr_{3})_{∗}((l_{a+b} × l_{j} × h^{0}) · μ) ∈ \overline{Ch}(X_{F_{q}}),

a contradiction. The statement for J′ is proved similarly. □

Lemma 81.11. The cycle β is F_{1}-rational. The cycles γ and γ_{1} are F_{q}-rational.

Proof. Since F_{1} = F (X), the cycle β is F_{1}-rational by definition.

Let pr_{23} : X^{3} → X^{2}, (x_{1}, x_{2}, x_{3}) ↦ (x_{2}, x_{3}) be the projection onto the product of the second and the third factors of X^{3}. The cycle l_{a+b} is F_{q}-rational; therefore γ = (pr_{23})_{∗}((l_{a+b} × h^{0} × h^{0}) · μ) is also F_{q}-rational. The cycle γ_{0} is F_{q}-rational as l_{a+b+c−1} is F_{q}-rational. It follows that γ_{1} is F_{q}-rational as well. □

Define

ξ(χ) := δ_{X^{2}}^{∗}(t_{12}^{∗}(χ) ∘ ( Sq_{X^{3}}^{2a}(χ) · (h^{0} × h^{0} × h^{c−a−1}))) for any χ ∈ Ch(X̄^{3}).

We must prove that the cycle ξ(μ) · (h^{0} × h^{c−a−1}) ∈ \overline{Ch}(X^{2}) contains h^{a+b} × l_{a+b} and does not contain h^{0} × l_{0}, i.e., we have to show that h^{a+b} × l_{b+c−1} ∈ ξ(μ) and h^{0} × l_{c−a−1} ∉ ξ(μ).

If h^{0} × l_{c−a−1} ∈ ξ(μ), then, passing from F to F_{1} = F (X), we have

l_{c−a−1} = (pr_{2})_{∗}((l_{0} × h^{0}) · ξ(μ)) ∈ \overline{Ch}(X_{F (X)}),

where pr_{2} : X^{2} → X is the projection onto the second factor of X^{2}, contradicting Corollary 72.6 as c − a − 1 ≥ a = 𝔦_{1}(X) = 𝔦_{0}(X_{F (X)}).

It remains to show that h^{a+b} × l_{b+c−1} ∈ ξ(μ). For any χ ∈ Ch(X̄^{2}), write coeff(χ) ∈ ℤ/2ℤ for the coefficient of h^{a+b} × l_{b+c−1} in χ. Since coeff(ν) = 0, it follows from (81.7) that

coeff (ξ(μ)) = coeff (ξ(h^{0} × β + h^{a+b} × γ)).

We claim that

(81.12)  coeff (ξ(h^{0} × β)) = 0 = coeff (ξ(h^{a+b} × γ)).

Indeed, since Sq_{X^{3}}^{2a}(h^{0} × β) = h^{0} × Sq_{X^{2}}^{2a}(β) by Theorem 61.14, we have

ξ(h^{0} × β) = h^{0} × δ_{X}^{∗}(β ∘ ( Sq_{X^{2}}^{2a}(β) · (h^{0} × h^{c−a−1})))

where δ_{X} : X → X^{2} is the diagonal morphism of X. Hence coeff (ξ(h^{0} × β)) = 0.

Since Sq_{X^{3}}^{2a}(h^{a+b} × γ) is h^{a+b} × Sq_{X^{2}}^{2a}(γ) plus terms having h^{j} with j > a + b as the first factor by Remark 79.2, we have

coeff(ξ(h^{a+b} × γ)) = coeff (h^{2a+2b} × δ_{X}^{∗}(γ ∘ ( Sq_{X^{2}}^{2a}(γ) · (h^{0} × h^{c−a−1})))) = 0.

This proves the claim.

It follows by claim (81.12) that

(81.13) coeff (ξ(μ)) = coeff (ξ(h^{0} × β + h^{a+b} × γ) − ξ(h^{0} × β) − ξ(h^{a+b} × γ)).

To compute the right-hand side in (81.13), we need only the terms h^{a+b}×Sq_{X^{2}}^{2a}(γ) in the formula for Sq_{X^{3}}^{2a}(h^{a+b} ×γ) since the other terms do not effect coeff. Therefore, we see that the right-hand side coefficient in (81.13) is equal to

coeff (h^{a+b} × δ_{X}^{∗}(γ ∘ ( Sq_{X^{2}}^{2a}(β) · (h^{0} × h^{c−a−1}))

+ β ∘ ( Sq_{X^{2}}^{2a}(γ) · (h^{0} × h^{c−a−1})))).

Consequently, to prove Proposition 81.3, it remains to prove

Lemma 81.14.

δ_{X}^{∗}(γ ∘ ( Sq_{X^{2}}^{2a}(β) · (h^{0} × h^{c−a−1})) + β ∘ ( Sq_{X^{2}}^{2a}(γ) · (h^{0} × h^{c−a−1}))) = l_{b+c−1}.

Proof. We start by showing that

(81.15)  δ_{X}^{∗}(β ∘ ( Sq_{X^{2}}^{2a}(γ) · (h^{0} × h^{c−a−1}))) = 0.

Note that Sq^{2a} vanishes on h^{0} × l_{a+b+c−1} by (81.4). Therefore Sq^{2a}(γ) = Sq^{2a}(γ_{1}) by (81.8). By Lemma 81.10 we may assume that dim X ≥ 4(a + b + c) − 2 (we shall need this assumption in order to apply Proposition 81.1), otherwise γ_{1} = 0.

Looking at the exponent of the first factor of the basis elements contained in Sq^{2a}(γ_{1}) and using Lemma 81.10, we see that none of the basis elements h^{j} × l_{j+b+c−1} and l_{j+b+c−1} × h^{j} with j < a + b + c are present in β ∘ ( Sq^{2a}(γ_{1}) · (h^{0} × h^{c−a−1})). As γ_{1} is F_{q}-rational by Lemma 81.11, Equation (81.15) holds by Proposition 81.1.

We compute Sq^{2a}(β_{0}) where β_{0} is as in (81.5). By Corollary 78.5 and Lemma 78.6, we have Sq^{0}(h^{a+b}) = h^{a+b}, Sq^{a}(h^{a+b}) = h^{2a+b}, and Sq^{j}(h^{a+b}) = 0 for all others, j ≤ 2a. Moreover, we have shown in (81.4) that Sq^{2a}(l_{a+b+c−1}) = 0. Therefore, Sq^{2a}(β_{0}) = Sym (h^{2a+b} × l_{b+c−1}) by Theorem 61.14.

Using Lemma 81.10, we have

γ_{0} ∘ ( Sq^{2a}(β_{0}) · (h^{0} × h^{c−a−1})) = l_{b+c−1} × h^{0}

and

(81.16)  δ_{X}^{∗}(γ_{0} ∘ ( Sq^{2a}(β_{0}) · (h^{0} × h^{c−a−1}))) = l_{b+c−1}.

The composition γ_{0} ∘ ( Sq^{2a}(β_{1}) · (h^{0} × h^{c−a−1})) is trivial. Indeed, by Lemma 81.10, every basis element of the cycle Sq^{2a}(β_{1}) · (h^{0} × h^{c−a−1}) has (as the second factor) either l_{j} with j ≥ 2a+b+c > 0 or h^{j} with j ≥ b+2c−1 > a+b+c−1, while the two basis elements of γ_{0} have h^{0} and l_{a+b+c−1} as the first factor. Consequently

(81.17)  δ_{X}^{∗}(γ_{0} ∘ ( Sq^{2a}(β_{1}) · (h^{0} × h^{c−a−1}))) = 0.

Looking at the exponent of the first factor of the basis elements contained in γ_{1} and using Lemma 81.10, we see that none of the basis elements h^{j} × l_{j+b+c−1} and l_{j+b+c−1} × h^{j} with j < a + b + c is present in γ_{1} ∘ ( Sq^{2a}(β) · (h^{0} × h^{c−a−1})). Therefore, the relation

(81.18)  δ_{X}^{∗}(γ_{1} ∘ ( Sq^{2a}(β) · (h^{0} × h^{c−a−1}))) = 0

holds by Proposition 81.1 in view of Lemma 81.11.

Taking the sum of the relations in (81.15)–(81.18), we have established the proof of Lemma 81.14. □

This completes the proof of Proposition 81.3. □

Theorem 81.2 is proved. □

Corollary 81.19 (cf. [75, Th. 1.1]). Let φ be an anisotropic quadratic form over a field of characteristic not 2. If 𝔥 = 𝔥(φ) > 1, then

v_{2}(𝔦_{1}) ≥ min (v_{2}(𝔦_{2}), …, v_{2}(𝔦_{𝔥})) − 1.

Proof. For any odd-dimensional φ, the statement is trivial, as all 𝔦_{q} are odd by Corollary 79.6. Assume that the inequality fails for an even-dimensional anisotropic φ. Note that in this case the difference

dim φ − 𝔦_{1} = 𝔦_{1} + 2(𝔦_{2} + ⋯ + 𝔦_{𝔥})

82. HOLES IN I^{n} 347

cannot be a power of 2 because it is larger than 2^{n} and not divisible by 2^{n+1} for n = v_{2}(𝔦_{1}). Therefore, by Corollary 80.10, the 1-primordial cycle on X^{2} does produce an integer. Therefore, the assumptions of Theorem 81.2 are satisfied, leading to a contradiction. □

Example 81.20. For an anisotropic quadratic form of dimension 6 and of trivial discriminant, we have 𝔥 = 2, 𝔦_{1} = 1, and 𝔦_{2} = 2. Therefore, the lower bound on v_{2}(𝔦_{1}) in Corollary 81.19 is exact.

82. Holes in I^{n}

Recall that F is a field of characteristic not 2. For every integer n ≥ 1, we set

dim I^{n}(F) := {dim φ | φ ∈ I^{n}(F) and anisotropic}.

and

dim I^{n} := ⋃ dim I^{n}(F),

where the union is taken over all fields F (of characteristic not 2).

In this section, we determine the set dim I^{n}. Theorem 82.8 states that dim I^{n} is the set of even nonnegative integers without the following disjoint open intervals (which we call holes in I^{n}):

U_{n−i} = (2^{n+1} − 2^{i+1}, 2^{n+1} − 2^{i}) , i = n, n − 1, …, 1.

The statement that U_{0} ∩ dim I^{n} = ∅ is the Hauptsatz (Theorem 23.7). This was originally proved by Arason and Pfister [10, Hauptsatz]. The statement for U_{1} ∩ dim I^{n} with n = 3 was originally proved in 1966 by Pfister [110, Satz 14], with n = 4 in 1998 by Hoffmann [55, Main Theorem], and for arbitrary n in 2000 by Vishik [133, Th. 6.4]. The statement that U_{n−i} ∩ dim I^{n} = ∅ for any n and i was conjectured by Vishik [133, Conj. 6.5]. A positive solution of the conjecture was announced by Vishik in 2002 but the proof is not available; a proof was given in [73].

Proposition 82.1. Let φ be a nonzero anisotropic form of even dimension with deg φ = n ≥ 1. If dim φ < 2^{n+1}, then dim φ = 2^{n+1} − 2^{i+1} for some i ∈ [0, n − 1].

Proof. We use the notation of §81. We prove the statement by induction on 𝔥 = 𝔥(φ). The case of 𝔥 = 1 is trivial.

So assume that 𝔥 > 1. As dim φ_{1} < dim φ < 2^{n+1} and deg φ_{1} = deg φ, where φ_{1} is the first (anisotropic) kernel form of φ, the induction hypothesis implies

dim φ_{1} = 2^{n+1} − 2^{i+1} for some i ∈ [1, n − 1].

Therefore, dim φ = 2^{n+1} − 2^{i+1} + 2𝔦_{1}. Since dim φ < 2^{n+1}, we have 𝔦_{1} < 2^{i}. In particular, v_{2}(dim φ − 𝔦_{1}) = v_{2}(𝔦_{1}). As 𝔦_{1} ≤ exp_{2} v_{2}(dim φ − 𝔦_{1}) by Proposition 79.4, it follows that 𝔦_{1} is a 2-power, say 𝔦_{1} = 2^{j} for some j ∈ [0, i − 1].

It follows by the induction hypothesis that each of the integers

dim φ_{1}, …, dim φ_{𝔥}

is divisible by 2^{i+1}. Therefore, v_{2}(𝔦_{q}) ≥ i for all q ∈ [2, 𝔥] and hence by Corollary 81.19, we have j ≥ i − 1. Consequently, j = i − 1, so dim φ = 2^{n+1} − 2^{i}. □

Corollary 82.2. Let φ be an anisotropic quadratic form such that φ ∈ I^{n}(F) for some n ≥ 1. If dim φ < 2^{n+1}, then dim φ = 2^{n+1} − 2^{i+1} for some i ∈ [0, n].

Proof. We may assume that φ ≠ 0. We have deg φ ≥ n by Corollary 25.12. Since 2^{deg φ} ≤ dim φ < 2^{n+1}, we must have deg φ = n. The result follows from Proposition 82.1. □

Corollary 82.3. Let φ ≠ 0 be an anisotropic quadratic form in I^{n}(F) with dim φ < 2^{n+1}. Then the higher Witt indices of φ are the successive 2-powers:

𝔦_{1} = 2^{i}, 𝔦_{2} = 2^{i+1}, …, 𝔦_{𝔥} = 2^{n−1},

where i = log_{2}(2^{n+1} − dim φ) − 1 is an integer.

Proof. By Corollary 82.2, we have dim φ = 2^{n+1} − 2^{i+1} for i as in the statement of Corollary 82.3, and dim φ_{1} = 2^{n+1} − 2^{j+1} for some j > i. It follows by Proposition 79.4 that 𝔦_{1} = 2^{i}. We proceed by induction on dim φ. □

We now show that every even value of dim φ for φ ∈ I^{n}(F) not forbidden by Corollary 82.2 is possible over some F. We start with some preliminary work.

Lemma 82.4. Let φ be a nonzero anisotropic quadratic form in I^{n}(F) and let dim φ < 2^{n+1} for some n ≥ 1. Then the 1-primordial cycle is the only primordial cycle in \overline{Ch}(X^{2}).

Proof. We induct on 𝔥 = 𝔥(φ). The case 𝔥 = 1 is trivial, so we assume that 𝔥 > 1. Let pr_{∗}^{2} : \overline{Ch}(X^{2}) → \overline{Ch}(X_{1}^{2}) be the homomorphism of Remark 72.5. Since the integer dim φ − 𝔦_{1} lies inside the open interval (2^{n}, 2^{n+1}), it is not a 2-power. Hence by Corollary 80.10, we have pr_{∗}^{2}(π) ≠ 0, where π ∈ \overline{Ch}(X^{2}) is the 1-primordial cycle. Therefore, by the induction hypothesis, the diagram of pr_{∗}^{2}(π) has points in every shell triangle. Thus, the diagram of π itself has points in every shell triangle. By Theorem 73.26, this means that π is the unique primordial cycle in \overline{Ch}(X^{2}). □

Corollary 82.5. Let φ be a nonzero anisotropic quadratic form in I^{n}(F) and let dim φ = 2^{n+1} − 2 for some n ≥ 1. Then for any i > 0, the group \overline{Ch}_{D+i}(X^{2}) contains no essential element.

Proof. By Lemma 82.4, the 1-primordial cycle is the only primordial cycle in \overline{Ch}(X^{2}). Since 𝔦_{1} = 1 by Corollary 82.3, we have dim π = D. To finish we apply Theorem 73.26. □

Lemma 82.6. Let F_{0} be a field (of char F_{0} ≠ 2) and let

F = F_{0}(t_{1j}, t_{2j})_{1≤j≤n}

be the field of rational functions in 2n variables. Then the quadratic form

⟨⟨t_{11}, …, t_{1n}⟩⟩′⊥ − ⟨⟨t_{21}, …, t_{2n}⟩⟩′

over F is anisotropic (where the prime denotes the pure subform of the Pfister form).

Proof. For any i = 0, 1, …, n, we set

φ_{i} = ⟨⟨t_{11}, …, t_{1i}⟩⟩ and ψ_{i} = ⟨⟨t_{21}, …, t_{2i}⟩⟩.

We prove that the form φ′_{i}⊥ − ψ′_{i} is anisotropic by induction on i. For i = 0 the statement is trivial. For i ≥ 1, we have:

φ′_{i}⊥ − ψ′_{i} ≃ (φ′_{i−1}⊥ − ψ′_{i−1})⊥ t_{1i}φ_{i−1}⊥ − t_{2i}ψ_{i−1}.

The summand φ′_{i−1}⊥ − ψ′_{i−1} is anisotropic by the induction hypothesis, while the forms φ_{i−1} and ψ_{i−1} are so by Corollary 19.6. Applying Lemma 19.5 repeatedly we conclude that the whole form is anisotropic. □

Define the anisotropic pattern of a quadratic form φ over F to be the set of integers dim(φ_{K})_{an} for all field extensions K/F. By Proposition 25.1, the anisotropic pattern of a form φ coincides with the set

{ dim φ − 2𝔧_{q}(φ) | q ∈ [0, 𝔥(φ)]}.

The following result is due to Vishik.

Proposition 82.7. Let F_{0} be a field (of char F_{0} ≠ 2) and let n ≥ 1 and m ≥ 2 be integers. Let

F = F_{0}(t_{i}, t_{ij})_{1≤i≤m, 1≤j≤n}

be the field of rational functions in variables t_{i} and t_{ij}. Then the anisotropic pattern of the quadratic form

φ = t_{1} · ⟨⟨t_{11}, …, t_{1n}⟩⟩⊥ … ⊥ t_{m} · ⟨⟨t_{m1}, …, t_{mn}⟩⟩

over F is the set

{2^{n+1} − 2^{i} | i ∈ [1, n + 1]} ∪ (2ℤ ∩ [2^{n+1}, m · 2^{n}]).

Proof. We first show that all the integers 2^{n+1} − 2^{i} are in the anisotropic pattern of φ. Indeed, the anisotropic part of φ over the field E obtained from F by adjoining the square roots of t_{31}, t_{41}, …, t_{m1}, of t_{1} and of −t_{2}, is isomorphic to the form

⟨⟨t_{11}, …, t_{1n}⟩⟩′⊥ − ⟨⟨t_{21}, …, t_{2n}⟩⟩′

of dimension 2^{n+1} − 2. This form is anisotropic by Lemma 82.6. The anisotropic pattern of this form is {2^{n+1} − 2^{i} | i ∈ [1, n + 1]} by Corollary 82.3.

Now assume that there is an even integer in the interval [2^{n+1}, m · 2^{n}] not in the anisotropic pattern of φ. Among all such integers take the smallest one and call it a. Let b = a − 2 and let c be the smallest integer greater than a and lying in the anisotropic pattern of φ. Let E be the field in the generic splitting tower of φ such that dim ψ = c where ψ = (φ_{E})_{an} and let Y be the projective quadric given by the quadratic form ψ. Let π ∈ \overline{Ch}(Y^{2}) be the 1-primordial cycle. We claim that

π = h^{0} × l_{𝔦_{1}−1} + l_{𝔦_{1}−1} × h^{0}

where 𝔦_{1} = 𝔦_{1}(Y). Indeed, since 𝔦_{1} = (c − b)/2 > 1 and 𝔦_{q}(Y) = 1 for all q such that dim ψ_{q} ∈ [2^{n+1} − 2, b − 2], the diagram of the cycle π does not have any point in the qth shell triangle for such q. For the integer q satisfying dim ψ_{q} = 2^{n+1} − 2, the cycle pr_{∗}^{2}(π) ∈ \overline{Ch}(Y_{q}^{2}) has dimension > dim Y_{q}, hence it is 0 by Corollary 82.5. The relation pr_{∗}^{2}(π) = 0 means that π has no point in any shell triangle with number > q.

It follows that π = h^{0} × l_{𝔦_{1}−1} + l_{𝔦_{1}−1} × h^{0}. By Corollary 80.8, the integer dim Y − 𝔦_{1} + 2 is a power of 2, say 2^{p}. Since

dim Y − 𝔦_{1} + 2 = (c − 2) − (c − b)/2 + 2 = (b + c)/2,

the integer 2^{p} lies inside the open interval (b, c). It follows that the integer 2^{p} satisfies 2^{n+1} ≤ 2^{p} < m·2^{n} and is not in the splitting pattern of the quadratic form φ. But every integer ≤ m · 2^{n} divisible by 2^{n} is evidently in the anisotropic pattern of φ. This contradiction establishes Proposition 82.7. □

Summarizing, we have Theorem 82.8. For any integer n ≥ 1,

dim I^{n} = {2^{n+1} − 2^{i} | i ∈ [1, n + 1]} ∪ (2ℤ ∩ [2^{n+1}, +∞)).

Proof. The inclusion ⊂ is given by Corollary 82.2, while the inclusion ⊃ follows by Proposition 82.7. □

Remark 82.9. The case of dimension 2^{n+1} − 2^{i} can be realized directly by the difference of two (i − 1)-linked n-fold Pfister forms (cf. Corollary 24.3).

83. On the 2-adic order of higher Witt indices, II

Throughout this section, X is an anisotropic quadric of dimension D over a field of characteristic not 2. We write 𝔦_{1}, …, 𝔦_{𝔥} and 𝔧_{1}, …, 𝔧_{𝔥} for the relative and absolute higher Witt indices of X, respectively, where 𝔥 is the height of X (cf. §81).

The main result of this section is Theorem 83.3 originally proved in [75]. It is used to establish further relations between higher Witt indices in Corollary 83.4.

We first establish further special properties of the 1-primordial cycle in addition to those in Proposition 73.27 and Theorem 81.2.

Lemma 83.1. Let π ∈ \overline{Ch}(X^{2}) be the 1-primordial cycle. Then Sq_{X^{2}}^{j}(π) = 0 for all j ∈ (0, 𝔦_{1}).

Proof. Let Sq = Sq_{X^{2}}. Assume that Sq^{j}(π) ≠ 0 for some j ∈ (0, 𝔦_{1}). By Remark 79.2, one sees that Sq^{j}(π) has a nontrivial intersection with an appropriate jth order derivative of π. As the derivative of π is minimal by Theorem 73.26, the cycle Sq^{j}(π) contains this derivative. It follows that Sq^{j}(π) has a point in the first left shell triangle, contradicting Lemma 79.3. □

Proposition 83.2. Let i be an integer such that h^{i} × l_{?} is contained in the 1primordial cycle. Then i is divisible by 2^{n+1} for any n ≥ 0 satisfying 𝔦_{1} > 2^{n}.

Proof. Assume that the statement is false. Let i be the minimal integer not divisible by 2^{n+1} and such that h^{i} × l_{?} is contained in the 1-primordial cycle π ∈ \overline{Ch}(X^{2}).

Note that π contains only essential basis elements and is symmetric. As dim π = D + 𝔦_{1} − 1, we have h^{i} × l_{i+𝔦_{1}−1} ∈ π.

For any nonnegative integer k divisible by 2^{n+1}, the binomial coefficient \binom{k}{l} with a nonnegative integer l is odd only if l is divisible by 2^{n+1} by Lemma 78.6. Therefore, Sq_{X}(h^{k}) = h^{k}(1 + h)^{k} is a sum of powers of h with exponents divisible by 2^{n+1}. It follows that the value Sq_{X}^{j}(π) contains the element Sq_{X}^{0}(h^{i}) × Sq_{X}^{j}(l_{i+𝔦_{1}−1}) = h^{i} ×Sq_{X}^{j}(l_{i+𝔦_{1}−1}) for any integer j. Since Sq_{X}^{j}(π) = 0 for j ∈ (0, 𝔦_{1}) by Lemma 83.1, we have

Sq_{X}^{j}(l_{i+𝔦_{1}−1}) = 0 for j ∈ (0, 𝔦_{1}).

Now look at the specific value Sq_{X}^{2^{v_{2}(i)}}(l_{i+𝔦_{1}−1}). Since i is not divisible by 2^{n+1} and 𝔦_{1} > 2^{n}, the degree 2^{v_{2}(i)} of the Steenrod operation lies in the interval (0, 𝔦_{1}). By Corollary 78.5, the value Sq_{X}^{2^{v_{2}(i)}}(l_{i+𝔦_{1}−1}) is equal to l_{i+𝔦_{1}−1−2^{v_{2}(i)}} multiplied by the binomial coefficient

\binom{D − i − 𝔦_{1} + 2}{2^{v_{2}(i)}}.

The integer D−𝔦_{1}+2 = dim φ−𝔦_{1} is divisible by 2^{n+1} by Proposition 79.4 as 𝔦_{1} > 2^{n}. Therefore the binomial coefficient is odd by Lemma 78.6. This is a contradiction, so it establishes the result. □

Theorem 83.3 (cf. [75, Th. 3.3]). Let X be an anisotropic quadric over a field of characteristic not 2. Suppose that the 1-primordial cycle π ∈ \overline{Ch}(X^{2}) produces the integer q. Then v_{2}(𝔦_{q}) ≥ v_{2}(𝔦_{1}).

Proof. Let n = v_{2}(𝔦_{1}). Then the integer 2^{n} divides dim φ − 𝔦_{1} by Proposition 79.4. Therefore, 2^{n} divides dim φ as well.

We have h^{𝔧_{q−1}} × l_{𝔧_{q−1}+𝔦_{1}−1} ∈ π by definition of q. Consequently, by Proposition 83.2, the integer 𝔧_{q−1} is divisible by 2^{n}. It follows that 2^{n} divides dim φ_{q−1} = dim φ − 2𝔧_{q−1}, where φ_{q−1} is the (q − 1)th (anisotropic) kernel of φ. If m < n for m = v_{2}(𝔦_{q}), then applying Proposition 79.4, we have 𝔦_{q} = 𝔦_{1}(φ_{q−1}) is equal to 2^{m} and, in particular, smaller than 𝔦_{1}. Therefore the 1-primordial cycle π has no points in the qth shell triangle. But the point h^{𝔧_{q−1}} × l_{𝔧_{q−1}+𝔦_{1}−1} ∈ π lies in the qth shell triangle. This contradiction establishes the theorem. □

Corollary 83.4 (cf. [75, Th. 1.1]). We have v_{2}(𝔦_{1}) ≤ max (v_{2}(𝔦_{2}), …, v_{2}(𝔦_{𝔥})) if the integer

dim φ − 𝔦_{1} = 𝔦_{1} + 2(𝔦_{2} + ⋯ + 𝔦_{𝔥})

is not a power of 2.

Proof. If the integer dim φ − 𝔦_{1} is not a 2-power, then the 1-primordial cycle does produce an integer by Corollary 80.10. The result follows by Theorem 83.3. □

84. Minimal height

Every nonnegative integer n is uniquely representable in the form of an alternating sum of 2-powers:

n = 2^{p_{0}} − 2^{p_{1}} + 2^{p_{2}} − ⋯ + (−1)^{r−1}2^{p_{r−1}} + (−1)^{r}2^{p_{r}}

for some integers p_{0}, p_{1}, …, p_{r} satisfying p_{0} > p_{1} > ⋯ > p_{r−1} > p_{r} + 1 > 0. We shall write P (n) for the set {p_{0}, p_{1}, …, p_{r}}. Note that p_{r} coincides with the 2-adic order v_{2}(n) of n. For n = 0 our representation is the empty sum, so P (0) = ∅.

Define the height 𝔥(n) of the integer n as the number of positive elements in P (n). Therefore, 𝔥(n) is the number |P (n)|, the cardinality of the set P (n) if n is even, while 𝔥(n) = |P (n)| − 1 if n is odd.

In this section we prove the following theorem conjectured by Rehmann and originally proved in [59]:

Theorem 84.1. Let φ be an anisotropic quadratic form over a field of characteristic not 2. Then

𝔥(φ) ≥ 𝔥(dim φ).

Remark 84.2. Let n ≥ 0 and let φ be an anisotropic excellent quadratic form of dimension n. It follows from Proposition 28.5 that 𝔥(φ) = 𝔥(n). Therefore, the bound in Theorem 84.1 is sharp.

We shall see (cf. Corollary 84.5) that Theorem 84.1 in odd dimensions is a consequence of Proposition 79.4. In even dimensions we shall also need Theorems 81.2 and 83.3. We shall also need several combinatorial arguments to prove Theorem 84.1.

Suppose φ is anisotropic. Let φ_{i} be the ith (anisotropic) kernel form of φ, and let n_{i} = dim φ_{i}, 0 ≤ i ≤ 𝔥(φ).

Lemma 84.3. For any i ∈ [1, 𝔥], the difference 𝔡(i) := 𝔥(n_{i−1}) − 𝔥(n_{i}) satisfies the following:

(I) If the dimension of φ is odd, then |𝔡(i)| = 1.

(II) If the dimension of φ is even, then |𝔡(i)| ≤ 2. Moreover, (+2) If 𝔡(i) = 2, then P (n_{i}) ⊂ P (n_{i−1}) and v_{2}(n_{i}) ≥ v_{2}(n_{i−1}) + 2. (+1) If 𝔡(i) = 1, the set difference P (n_{i}) \ P (n_{i−1}) is either empty or

consists of a single element p, in which case both integers p − 1 and p + 1 lie in P (n_{i−1}).

(0) If 𝔡(i) = 0, the set difference P (n_{i}) \ P (n_{i−1}) consists of one element p and either p − 1 or p + 1 lies in P (n_{i−1}).

(−1) If 𝔡(i) = −1, the set difference P (n_{i}) \ P (n_{i−1}) consists either of two elements p − 1 and p + 1 for some p ∈ P (n_{i−1}) or the set difference consists of one element.

(−2) If 𝔡(i) = −2, the set difference P (n_{i}) \ P (n_{i−1}) consists of two elements, i.e., P (n_{i}) ⊃ P (n_{i−1}). Moreover, in this case one of these two elements is equal to p + 1 for some p ∈ P (n_{i−1}).

Proof. Write p_{0}, p_{1}, …, p_{r} for the elements of P (n_{i−1}) in descending order. We have n_{i} = n_{i−1} − 2𝔦_{i}. We also know by Proposition 79.4 that there exists a nonnegative integer m such that 2^{m} < n_{i−1}, 𝔦_{i} ≡ n_{i−1} (mod 2^{m}), and 1 ≤ 𝔦_{i} ≤ 2^{m}. The condition 2^{m} < n_{i−1} implies m < p_{0}. Let p_{s} be the element with maximal even s satisfying m < p_{s}.

If m = p_{s} − 1, then 𝔦_{i} = 2^{p_{s}−1} − 2^{p_{s+1}} + 2^{p_{s+2}} − … and, therefore,

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ − 2^{p_{s−1}} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r−1}2^{p_{r}}.

If s = r and p_{r−1} + 1 = p_{r−2}, then P (n_{i}) equals P (n_{i−1}) without p_{r−2} and p_{r}. Otherwise, P (n_{i}) equals P (n_{i−1}) without p_{s}.

Therefore, we may assume that m < p_{s} − 1.

If s = r, then 𝔦_{i} = 2^{m} and n_{i} = n_{i−1} − 2^{m+1}. If m = p_{r} − 2, we have P (n_{i}) is obtained from P (n_{i−1}) by replacing p_{r} with p_{r} − 1. If m < p_{r} − 2, we have P (n_{i}) equals P (n_{i−1}) with m + 1 added.

Therefore, we may assume in addition that s < r.

If p_{s} − 1 > m > p_{s+1}, then 𝔦_{i} = 2^{m} − 2^{p_{s+1}} + 2^{p_{s+2}} − … and, therefore,

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ − 2^{p_{s−1}} + 2^{p_{s}} − 2^{m+1} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r+1}2^{p_{r}}.

This is the correct representation of n_{i} and, therefore, P (n_{i}) equals P (n_{i−1}) with m + 1 added.

It remains to consider the case with m ≤ p_{s+1} while s < r. In this case, first assume that s = r − 1. Then 𝔦_{i} = 2^{m} and n_{i} = n_{i−1} − 2^{m+1}.

If m < p_{r} − 2, then P (n_{i}) equals P (n_{i−1}) with p_{r} + 1 and m + 1 added.

If m = p_{r} −2, then P (n_{i}) equals P (n_{i−1}) with p_{r} removed and p_{r} +1 and p_{r} −1 added.

If m = p_{r} − 1, one has two possibilities. If p_{r−1} > p_{r} + 2, then P (n_{i}) equals P (n_{i−1}) with p_{r} removed and p_{r} + 1 added. If p_{r−1} = p_{r} + 2, then P (n_{i}) equals P (n_{i−1}) with p_{r} and p_{r−1} removed while p_{r} + 1 added.

Finally, if m = p_{r}, then either p_{r−1} = p_{r} + 2 and P (n_{i}) equals P (n_{i−1}) without p_{r−1}, or P (n_{i}) equals P (n_{i−1}) with p_{r} + 2 added.

To finish the proof we may assume that m ≤ p_{s+1} and s < r − 1. We have: 𝔦_{i} = 2^{p_{s+2}} − 2^{p_{s+3}} + ⋯ + (−1)^{r}2^{p_{r}} and

n_{i} = 2^{p_{0}} − 2^{p_{1}} + ⋯ + 2^{p_{s}} − 2^{p_{s+1}+1} + 2^{p_{s+1}} − 2^{p_{s+2}} + ⋯ + (−1)^{r+1}2^{p_{r}}.

Thus, if p_{s} > p_{s+1} + 1, then P (n_{i}) equals P (n_{i−1}) with p_{s+1} + 1 added; otherwise P (n_{i}) is P (n_{i−1}) with p_{s} removed. □

Corollary 84.4. Let φ be an anisotropic odd-dimensional quadratic form and let i ∈ [1, 𝔥(φ)]. Then

𝔥(n_{i−1}) − 𝔥(n_{i}) ≤ 1.

Corollary 84.5. Let φ be an anisotropic quadratic form of odd dimension n. Then 𝔥(φ) ≥ 𝔥(n).

Proof. Let 𝔥 := 𝔥(φ). As dim φ is odd, n_{𝔥} = 1. Then 𝔥(n_{𝔥}) = 0 and by Corollary 84.4, we have 𝔥(n_{i−1}) − 𝔥(n_{i}) ≤ 1 for every i ∈ [1, 𝔥]. Therefore, 𝔥(n_{0}) ≤ 𝔥. Since φ is anisotropic, n = n_{0}, and the result follows. □

Remark 84.6. By Lemma 84.3, for any quadratic form φ of odd dimension n, we have 𝔥(n_{i}) = 𝔥(n_{i−1}) ± 1. Therefore 𝔥(φ) ≡ 𝔥(n) (mod 2).

Proposition 84.7. Let φ be an anisotropic quadratic form of even dimension n and 𝔥 := 𝔥(φ). Suppose that v_{2}(n_{i}) ≥ v_{2}(n_{i−1}) + 2 for some i ∈ [1, 𝔥). Then the open interval (i, 𝔥) contains an integer i′ satisfying |v_{2}(n_{i′}) − v_{2}(n_{i−1})| ≤ 1.

Proof. It suffices to consider the case i = 1. Note that 𝔥 ≥ 2. Set p = v_{2}(n_{0}). By assumption, we have v_{2}(n_{1}) ≥ p + 2. Therefore, v_{2}(𝔦_{1}) = p − 1. Clearly, the integer n_{0} − 𝔦_{1} = 𝔦_{1} + n_{1} is not a power of 2. Therefore, by Corollary 80.10, the 1-primordial cycle of \overline{Ch}(X^{2}) produces an integer j ∈ [2, 𝔥]. We shall show that either v_{2}(n_{j−1}) or v_{2}(n_{j}) lies in [p − 1, p + 1] for this j. We then take i′ = j − 1 in the first case and i′ = j in the second case. Note that i′ ≠ 1 and i′ ≠ 𝔥 as v_{2}(n_{1}) ≥ p + 2, while v_{2}(n_{𝔥}) = ∞.

By Theorem 83.3, we have v_{2}(𝔦_{j}) ≥ p − 1. Consequently, v_{2}(n_{j−1}) ≥ p − 1 by Proposition 79.4 as well. Since n_{1} = 2(𝔦_{2} + ⋯ + 𝔦_{j−1}) + n_{j−1}, it follows that v_{2}(𝔦_{2} + ⋯ + 𝔦_{j−1}) + 1 ≥ p − 1. If v_{2}(𝔦_{2} + ⋯ + 𝔦_{j−1}) < p + 1, then v_{2}(n_{j−1}) = v_{2}(𝔦_{2}+⋯+𝔦_{j−1})+1 ∈ [p−1, p+1]. So, we may assume that v_{2}(𝔦_{2}+⋯+𝔦_{j−1}) ≥ p+1 and apply Theorem 81.2 to conclude that v_{2}(𝔦_{j}) ≤ p. We have v_{2}(𝔦_{j}) ∈ {p−1, p}. If v_{2}(n_{j−1}) > v_{2}(𝔦_{j})+1, then v_{2}(n_{j}) = v_{2}(𝔦_{j})+1 ∈ {p, p+1}. If v_{2}(n_{j−1}) = v_{2}(𝔦_{j})+1, then v_{2}(n_{j−1}) ∈ {p, p + 1}. Finally, if v_{2}(n_{j−1}) < v_{2}(𝔦_{j}) + 1, then v_{2}(n_{j−1}) = v_{2}(𝔦_{j}), hence v_{2}(n_{j−1}) ∈ {p − 1, p}. □

Corollary 84.8. Let φ be an anisotropic quadratic form of even dimension n. Suppose that v_{2}(n_{i}) ≥ v_{2}(n_{i−1})+2 for some i ∈ [1, 𝔥). Set p = v_{2}(n_{i−1}). Then there exists i′ ∈ (i, 𝔥) such that the set P (n_{i′}) contains an element p′ with |p′ − p| ≤ 1.

Proof. Let i′ be the integer in the conclusion of Proposition 84.7. Then p′ = v_{2}(n_{i′}) works. □

We now prove Theorem 84.1.

Proof of Theorem 84.1. By Corollary 84.5, we need only prove Theorem 84.1 for even-dimensional forms. Let 𝔥 := 𝔥(φ) ≥ 1 and let {n_{0} > n_{1} > ⋯ > n_{𝔥}} with n_{i} = dim φ_{i} be the anisotropic pattern of φ with n = n_{0} even.

Let H be the set {1, 2, …, 𝔥}. For any i ∈ H, let 𝔡(i) := 𝔥(n_{i−1}) − 𝔥(n_{i}). Recall that 𝔡(i) ≤ 2 for any i ∈ H by Lemma 84.3. Let C be the subset of H consisting of all those i ∈ H such that 𝔡(i) = 2. We shall construct a map f : C → H satisfying 𝔡(j) ≤ 1 − |f^{−1}(j)| for any j ∈ f(C). In particular, we shall have f(C) ⊂ H \ C. Once such a map is constructed, we establish Theorem 84.1 as follows: The subsets f^{−1}(j) ∪ {j} ⊂ H, where j runs over H \ C, are disjoint and cover H. In addition, the average value of 𝔡 on each such subset is ≤ 1, so the average value ( ∑_{i∈H} 𝔡(i))/𝔥 = 𝔥(n)/𝔥 of 𝔡 on H is ≤ 1, i.e., 𝔥(n) ≤ 𝔥.

So it remains to define the map f with the desired properties. Let i ∈ C. By Lemma 84.3, we have v_{2}(n_{i}) ≥ v_{2}(n_{i−1}) + 2. Therefore, by Corollary 84.8, there exists i′ ∈ (i, 𝔥) such that the set P (n_{i′}) contains an element p′ satisfying |p′ −p| ≤ 1 for p = v_{2}(n_{i−1}). Taking the minimal i′ with this property, set f(i) = i′. We also define g(i) to be the minimal element of P (n_{f(i)}) satisfying |g(i) − p| ≤ 1.

This defines the map f. To finish, we must show that f has the desired property. First observe that by the definition of f, for any i ∈ C and any j ∈ [i, f(i) − 1]

the set P (n_{j}) does not contain any element p with |p − v_{2}(n_{i−1})| ≤ 1. It follows that if f(i_{1}) = f(i_{2}) for some i_{1} ≠ i_{2}, then for p_{1} = v_{2}(n_{i_{1}−1}) with p_{2} = v_{2}(n_{i_{2}−1}) one has |p_{2} − p_{1}| ≥ 2. Moreover, if g(i_{1}) = g(i_{2}), then |p_{1} − p| ≤ 1 and |p_{2} − p| ≤ 1 for p = g(i_{1}) = g(i_{2}) by definition of g. Therefore, we have

(84.9) if f(i_{1}) = f(i_{2}) and g(i_{1}) = g(i_{2}) for some i_{1} ≠ i_{2},

then |p_{2} − p_{1}| = 2 for p_{1} = v_{2}(n_{i_{1}−1}) and p_{2} = v_{2}(n_{i_{2}−1}).

Let j ∈ f(C). By the definition of f, the set difference P (n_{j}) \ P (n_{j−1}) is nonempty. Then 𝔡(j) ≠ 2 by Lemma 84.3(II)(+2). Moreover, the above set difference contains an element p satisfying {p − 1, p + 1} ⊄ P (n_{j−1}). Consequently, 𝔡(j) ≠ 1 by Lemma 84.3(II)(+1). Therefore, 𝔡(j) ≤ 0 by Lemma 84.3.

Now let j be an element of f(C) with |f^{−1}(j)| ≥ 2. Let i_{1} < i_{2} be two different elements of f^{−1}(j). Note that i_{1} < i_{2} < j. Moreover, if p_{1} = v_{2}(n_{i_{1}−1}) and p_{2} = v_{2}(n_{i_{2}−1}), then, by the definition of f(i_{1}), we have |p_{2} − p_{1}| > 1. We shall show that 𝔡(j) ≤ −1. We already know that 𝔡(j) ≤ 0. If 𝔡(j) = 0, then by Lemma 84.3(II)(−0), the set difference P (n_{j})\P (n_{j−1}) consists of one element p′ and either p′ − 1 or p′ + 1 lies in P (n_{j−1}). Since the difference P (n_{j}) \ P (n_{j−1}) consists of one element p′, we have p′ = g(i_{1}) = g(i_{2}). It follows that {p_{1}, p_{2}} = {p′ − 1, p′ + 1}. Consequently, the set P (n_{j−1}) contains neither p′ − 1 nor p′ + 1, a contradiction. Thus we have proved that 𝔡(j) ≤ −1 if |f^{−1}(j)| ≥ 2.

Now let j be an element of f(C) with |f^{−1}(j)| ≥ 3. Let i_{1}, i_{2}, i_{3} be three different elements of f^{−1}(j). The equalities g(i_{1}) = g(i_{2}) = g(i_{3}) cannot take place simultaneously, as otherwise, by (84.9), we would have |p_{2} − p_{1}| = 2, |p_{3} − p_{2}| = 2, and |p_{1} − p_{3}| = 2, a contradiction. However, the set difference P (n_{j}) \ P (n_{j−1}) can have at most two elements. Therefore, we may assume that g(i_{1}) = g(i_{2}) and that g(i_{3}) is different from g(i_{1}) = g(i_{2}). Set p′ = g(i_{1}) = g(i_{2}). We shall show that 𝔡(j) = −2. We already know that 𝔡(j) ≤ −1. If 𝔡(j) = −1, then by Lemma 84.3(II)(−1), the set difference P (n_{j}) \ P (n_{j−1}) consists of p̃− 1 and p̃+ 1 for some p̃ ∈ P (n_{j−1}). However, p′ is neither p̃ − 1 nor p̃ + 1, a contradiction.

We finish the proof by showing that |f^{−1}(j)| is never ≥ 4. Indeed, if |f^{−1}(j)| ≥ 4, then the set difference P (n_{j}) \ P (n_{j−1}) contains two elements p′ and p′′ with none of p′ ± 1 or p′′ ± 1 lying in P (n_{j−1}), contradicting Lemma 84.3. □

CHAPTER XVI The Variety of Maximal Totally Isotropic Subspaces

So far in this book, the projective quadric has been the only variety associated with a quadratic form. In this chapter, we introduce another variety, the variety of maximal isotropic subspaces.

85. The variety Gr(φ)

Let φ be a nondegenerate quadratic form on V over F. In this chapter, we study the scheme Gr(φ) of maximal totally isotropic subspaces of V. We view Gr(φ) as a closed subscheme of the Grassmannian variety of V. Let n be the integer part of (dim φ − 1)/2, so dim φ = 2n + 1 or 2n + 2.

Example 85.1. If dim φ = 1, we have Gr(φ) = Spec F. If dim φ = 2 or 3, then Gr(φ) coincides with the quadric of φ; that is Gr(φ) = Spec C_{0}(φ) if dim φ = 2 and Gr(φ) is the conic curve associated to the quaternion algebra C_{0}(φ) if dim φ = 3.

The orthogonal algebraic group O(V, φ) acts transitively on Gr(φ). We write O^{+}(V, φ) for the (connected) special orthogonal group (cf. [86, §23]). If dim φ is odd, O^{+}(V, φ) acts transitively on Gr(φ) and therefore, Gr(φ) is a smooth projective variety over F.

Suppose that dim φ = 2n+2 is even. Then the group O(V, φ) has two connected components, one is O^{+}(V, φ). The factor group O(V, φ)/ O^{+}(V, φ) is identified with the Galois group over F of the center Z of the even Clifford algebra C_{0}(V, φ). Recall that Z is an étale quadratic F -algebra (cf. Proposition 11.6). The class of Z(φ) in Ét_{2}(F) is the discriminant of φ (cf. §13).

A point of Gr(φ) over a commutative ring R is a totally isotropic direct summand P of rank n + 1 of the R-module V_{R} = V ⊗_{F} R. Since p^{2} = 0 in the Clifford algebra C(V, φ)_{R} for every p ∈ P, the inclusion of P into V_{R} gives rise to an injective R-module homomorphism h : Λ^{n+1}(P) → C(V, φ)_{R}. Let W be the image of h. Since ZW = W, left multiplication by elements of the center Z of C_{0}(V, φ) defines an F -algebra homomorphism Z → End_{R}(W) = R. Therefore, we have a morphism Gr(φ) → Spec Z, so Gr(φ) is a scheme over Z.

If the discriminant of φ is trivial, i.e., Z = F × F, the scheme Gr(φ) has two smooth (irreducible) connected components Gr_{1} and Gr_{2} permuted by the quotient O(V, φ)/ O^{+}(V, φ). More precisely, they are isomorphic under any reflection of V. If Z is a field, the discriminant of φ_{Z} is trivial and therefore Gr(φ) is isomorphic to a connected component of Gr(φ_{Z}).

The varieties of even- and odd-dimensional forms are related by the following statement.

Proposition 85.2. Let φ be a nondegenerate quadratic form on V over F of dimension 2n + 2 and trivial discriminant and let φ′ be a nondegenerate subform of φ on a subspace V′ ⊂ V of codimension 1. Let Gr_{1} be a connected component of Gr(φ). Then the assignment U ↦ U ∩ V′ gives rise to an isomorphism

Gr_{1} →^{∼} Gr(φ′).

Proof. Since both varieties Gr_{1} and Gr(φ′) are smooth, it suffices to show that the assignment induces a bijection on points over any field extension L/F. Moreover, we may assume that L = F. Let U′ ⊂ V′ be a totally isotropic subspace of dimension n. Then the orthogonal complement U′^{⊥} of U′ in V is (n+2)-dimensional and the induced quadratic form on H = U′^{⊥}/U′ has trivial discriminant (i.e., H is a hyperbolic plane). The space H has exactly two isotropic lines permuted by a reflection. Therefore, the pre-images of these lines in V are two totally isotropic subspaces of dimension n + 1 living in different components of Gr(φ). Thus exactly one of them represents a point of Gr_{1} over F. □

Let φ′ be a nondegenerate subform of codimension 1 of a nondegenerate quadratic form φ of even dimension. Let Z be the discriminant of φ. By Proposition 85.2, we have that Gr(φ′)_{Z} is isomorphic to a connected component Gr_{1} of Gr(φ_{Z}) and therefore, Gr(φ) ≃ Gr_{1} ≃ Gr(φ′)_{Z}.

Example 85.3. If dim φ = 4, then Gr(φ) is the conic curve (over Z) associated to the quaternion algebra C_{0}(φ).

Exercise 85.4. Show that if 3 ≤ dim φ ≤ 6, then Gr(φ) is isomorphic to the Severi-Brauer variety associated to the even Clifford algebra C_{0}(φ).

86. The Chow ring of Gr(φ) in the split case

In this section, we present the calculation of the Chow groups of Gr(φ) given by Vishik in [134]. Set Gr := Gr(φ) and r := dim φ−n−1, where n is the integer part of (dim φ − 1)/2. We define the tautological vector bundle E over Gr of rank r to be the restriction of the tautological vector bundle over the Grassmannian variety of V, i.e., variety E is the closed subvariety of the trivial bundle V 𝟙 := V × Gr consisting of pairs (u, U) such that u ∈ U. The projective bundle ℙ(E) is a closed subvariety of X × Gr, where X is the (smooth) projective quadric of φ.

Let E^{⊥} be the kernel of the natural morphism V 𝟙 → E^{∨} given by the polar bilinear form 𝔟_{φ}. If dim φ = 2n + 2, we have U^{⊥} = U for any totally isotropic subspace U ⊂ V of dimension n + 1, hence E^{⊥} = E.

In the case when dim φ = 2n + 1, the situation is as follows. For any totally isotropic subspace U ⊂ V of dimension n, the orthogonal complement U^{⊥} contains U as a subspace of codimension 1. Therefore, E^{⊥} is a vector bundle over Gr of rank n + 1 containing E. The fiber of E^{⊥} over U is the orthogonal complement U^{⊥}.

Now suppose that φ is isotropic. Choose an isotropic line L ⊂ V. Set Ṽ = L^{⊥}/L and let φ̃ be the quadratic form on Ṽ induced by φ. Denote the projective quadric of φ̃ by X̃.

A totally isotropic subspace of Ṽ of dimension r − 1 is of the form U/L, where U is a totally isotropic subspace of V of dimension r containing L. Therefore, we can view the variety G̃r := Gr(φ̃) of maximal totally isotropic subspaces of Ṽ as a closed subvariety of Gr. Denote by i : G̃r → Gr the closed embedding.

86. THE CHOW RING OF Gr(φ) IN THE SPLIT CASE 357

Let U be a totally isotropic subspace of V of dimension r that does not contain L. Then dim(U ∩ L^{⊥}) = r − 1 and ((U ∩ L^{⊥}) + L)/L is a totally isotropic subspace of Ṽ of dimension r − 1.

Lemma 86.1. The morphism f : Gr \G̃r → G̃r that takes U to ((U ∩ L^{⊥}) + L)/L is an affine bundle.

Proof. We use the criterion of Lemma 52.12. Let R be a local commutative F -algebra. An F -morphism Spec R → G̃r, or equivalently an R-point of G̃r, is given by a direct summand Ũ of the R-module V_{R} = V ⊗_{F} R of rank r with L_{R} ⊂ Ũ.

An R-point of Gr \G̃r is a totally isotropic direct summand U of V_{R} of rank r with U + L_{R} a direct summand U of V_{R} of rank r + 1. If f(U) = Ũ/L_{R}, then Ũ ⊂ U+L_{R}. The assignment U ↦ (U+L_{R})/Ũ gives rise to an isomorphism between the fiber Spec R ×G̃r (Gr \G̃r) of f over Ũ/L_{R} and ℙ_{R}(V_{R}/Ũ) \ ℙ_{R}(L_{R}^{⊥}/Ũ) ≃ 𝔸_{R}^{n}. By Lemma 52.12, we have f is an affine bundle. □

Note that dim Gr = dim G̃r + n, so dim Gr = n(n + 1)/2.

By Lemma 86.1, Gr is a cellular variety with the short filtration G̃r ⊂ Gr, hence by Corollary 66.4, we have a decomposition of Chow motives:

(86.2)  M(Gr) = M(G̃r) ⊕ M(G̃r)(n).

The morphism M(G̃r) → M(Gr) is induced by the embedding i : G̃r → Gr and the morphism M(G̃r)(n) → M(Gr) is given by the transpose of the closure of the graph of f, the class of which we shall denote by β ∈ CH(G̃r × Gr). The correspondence β is given by the scheme of pairs (W/L, U), where U is a totally isotropic r-dimensional subspace of V, W is a totally isotropic r-dimensional subspace of L^{⊥} containing L, and dim(U + W) ≤ r + 1.

For the rest of this section, we shall assume that φ is split. It follows by induction, using (86.2) and Example 85.1, that CH(Gr) is a free abelian group of rank 2^{r}. We shall determine the multiplicative structure of CH(Gr).

Since the motive of X (and also Gr) is split, we have CH(X × Gr) = CH(X) ⊗ CH(Gr) by Proposition 64.3. In other words, CH(X × Gr) is a free module over CH(Gr) with basis

{h^{k} × [Gr], l_{k} × [Gr] | k ∈ [0, n − 1]} if dim φ = 2n + 1,
{h^{k} × [Gr], l_{k} × [Gr], l_{n} × [Gr], l′_{n} × [Gr] | k ∈ [0, n − 1]} if dim φ = 2n + 2.

Note that in the even-dimensional case, we have assumed that X is oriented.

In both cases, ℙ(E) is a closed subvariety of X×Gr of codimension n. Therefore, in the odd-dimensional case there are unique elements e_{k} ∈ CH^{k}(Gr), k ∈ [0, n], satisfying

(86.3)  [ℙ(E)] = l_{n−1} × e_{0} + ∑_{k=1}^{n} h^{n−k} × e_{k}

in CH(X×Gr). Pulling this back with respect to the canonical morphism X_{F (Gr)} → X × Gr, we see that e_{0} = 1.

In the even-dimensional case, there are unique elements e_{k} ∈ CH^{k}(Gr), k ∈ [0, n] and e′_{0} ∈ CH^{0}(Gr), satisfying

(86.4)  [ℙ(E)] = l_{n} × e_{0} + l′_{n} × e′_{0} + ∑_{k=1}^{n} h^{n−k} × e_{k}

in CH(X × Gr). Choose a totally isotropic subspace U ⊂ V of dimension n + 1 so that [ℙ(U)] = l_{n} in CH(X) and let U′ be a reflection of U. It follows from Exercise 68.4 that [ℙ(U′)] = l′_{n}. Let g denote the generic point of Gr whose closure contains [U]. Let g′ be another generic point of Gr whose closure contains [U′]. Note that CH^{0}(Gr) = ℤ[g] ⊕ ℤ[g′]. Pulling back equation (86.4) with respect to the two morphisms X → X × Gr given by the points [U] and [U′] respectively, we see that e_{0} = [g] and e′_{0} = [g′]. In particular, e_{0} and e′_{0} are orthogonal idempotents of CH^{0}(Gr), hence e_{0} + e′_{0} = 1.

It follows that for every totally isotropic subspace W ⊂ V of dimension n + 1 with [W] in the closure of g (resp. g′), we have [W] = l_{n} (respectively [W] = l′_{n}). In particular, to give an orientation of X is to choose one of the two connected components of Gr.

The multiplication rule in Proposition 68.1 for CH(X) implies that in both cases

e_{k} = p_{∗}((l_{n−k} × 1) · [ℙ(E)])

for k ∈ [1, n], where p : X × Gr → Gr is the projection.

We view the cycle γ = [ℙ(E)] in CH(X × Gr) as the incidence correspondence X ⇝ Gr. It follows from Proposition 63.2 that the induced homomorphism γ_{∗} : CH(X) → CH(Gr) takes l_{n−k} to e_{k}.

Let s : ℙ(E) → Gr and t : ℙ(E) → X be the two projections. Proposition 62.7 provides the following simple formula for e_{k}:

(86.5)  e_{k} = s_{∗} ∘ t^{∗}(l_{n−k}).

Lemma 86.6. We have e_{n} = [G̃r] in CH^{n}(Gr).

Proof. The element t^{∗}(l_{0}) coincides with the cycle of the intersection ({L} × Gr) ∩ ℙ(E) = {L} × G̃r. It follows from (86.5) that [G̃r] = s_{∗} ∘ t^{∗}(l_{0}) = e_{n}. □

We write h̃ and l̃_{i} for the standard generators of CH(X̃). Recall that the incidence correspondence α : X̃ ⇝ X is given by the scheme of pairs (A/L, B) of one-dimensional isotropic subspaces of Ṽ and V respectively with B ⊂ A. By Lemma 72.3, we can orient X̃ (in the case dim φ is even) so that α_{∗}(l̃_{k−1}) = l_{k} and α^{∗}(l_{k}) = l̃_{k−1} for all k.

Denote by ẽ_{k} ∈ CH^{k}(G̃r) the elements given by (86.3) or (86.4) for G̃r. Similarly, we have the incidence correspondence γ̃ : X̃ ⇝ G̃r with γ̃_{∗}(l̃_{n−1−k}) = ẽ_{k}.

Lemma 86.7. The diagram of correspondences

[figure: X̃; γ̃; α; X; γ; α^{t}; X̃; γ̃; G̃r; β; Gr; i^{t}; G̃r]

is commutative.

Proof. By Corollary 57.22, all calculations can be done on the level of cycles representing the correspondences. By definition of the composition of correspondences, the compositions γ ∘ α and β ∘ γ̃ coincide with the cycle of the subscheme of X̃ × Gr consisting of all pairs (A/L, U) with dim(A + U) ≤ r + 1. Similarly, the compositions γ̃ ∘ α^{t} and i^{t} ∘ γ coincide with the cycle of the subscheme of X × G̃r consisting of all pairs (B, Ũ/L) with B ⊂ Ũ. □

Corollary 86.8. We have β_{∗}(ẽ_{k}) = e_{k} and i^{∗}(e_{k}) = ẽ_{k} for all k ∈ [0, n − 1].

Proof. The equalities β_{∗}(ẽ_{0}) = e_{0} and i^{∗}(e_{0}) = ẽ_{0} follow from the fact that X and X̃ have compatible orientations. If k ≥ 1, we have by Lemma 86.7,

β_{∗}(ẽ_{k}) = β_{∗} ∘ γ̃_{∗}(l̃_{n−1−k}) = γ_{∗} ∘ α_{∗}(l̃_{n−1−k}) = γ_{∗}(l_{n−k}) = e_{k},

and

i^{∗}(e_{k}) = i_{∗}^{t}(e_{k}) = i_{∗}^{t} ∘ γ_{∗}(l_{n−k})
= γ̃_{∗} ∘ α_{∗}^{t}(l_{n−k}) = γ̃_{∗} ∘ α^{∗}(l_{n−k}) = γ̃_{∗}(l̃_{n−1−k}) = ẽ_{k}.  □

For a subset I of [0, n] let e_{I} be the product of e_{k} for all k ∈ I. Similarly, we define ẽ_{J} for any subset J ⊂ [0, n − 1].

Corollary 86.9. We have i_{∗}(ẽ_{J}) = e_{J} · e_{n} = e_{J∪{n}} for every J ⊂ [0, n − 1].

Proof. By Corollary 86.8, we have i^{∗}(e_{J}) = ẽ_{J}. It follows from Lemma 86.6 and the Projection Formula (Proposition 56.9) that

i_{∗}(ẽ_{J}) = i_{∗}(i^{∗}(e_{J}) · 1) = e_{J} · i_{∗}(1) = e_{J} · e_{n} = e_{J∪{n}}.  □

Corollary 86.10. The monomial e_{[0, n]} = e_{0}e_{1} ⋯ e_{n} is the class of a rational point in CH_{0}(Gr).

Proof. The statement follows from the formula e_{[0, n]} = i_{∗}(ẽ_{[0, n−1]}) and by induction on n. □

Let j : Gr \G̃r → Gr be the open embedding. Let f : Gr \G̃r → G̃r be the morphism in Lemma 86.1.

Lemma 86.11. We have f^{∗}(ẽ_{J}) = j^{∗}(e_{J}) for any J ⊂ [0, n − 1].

Proof. It suffices to prove that f^{∗}(ẽ_{k}) = j^{∗}(e_{k}) for all k ∈ [0, n − 1]. By the construction of β (cf. §66), we have β^{t} ∘ j = f. It follows from Corollary 86.8 that f^{∗}(ẽ_{k}) = j^{∗} ∘ (β^{t})^{∗}(ẽ_{k}) = j^{∗} ∘ β_{∗}(ẽ_{k}) = j^{∗}(e_{k}). □

Theorem 86.12. Let φ be a nondegenerate quadratic form on V over F of dimension 2n + 1 or 2n + 2 and let Gr be the scheme of maximal totally isotropic subspaces of V. Then the set of monomials e_{I} for all 2^{n} subsets I ⊂ [1, n] is a basis of CH(Gr) over CH^{0}(Gr).

Proof. We induct on n. The localization property gives the exact sequence (cf. the proof of Theorem 66.2)

0 → CH(G̃r) ⟶^{i_{∗}} CH(Gr) ⟶^{j^{∗}} CH(Gr \G̃r) → 0.

By the induction hypothesis and Corollary 86.9, the set of monomials e_{I} for all I containing n is a basis of the image of i_{∗}. Since f^{∗} : CH(G̃r) → CH(Gr \G̃r) is an isomorphism by Theorem 52.13, again by the induction hypothesis and Lemma 86.11, the set of all the elements j^{∗}(e_{I}) with n ∉ I is a basis of CH(Gr \G̃r). The statement follows. □

We now can compute the Chern classes of the tautological vector bundle E over Gr.

Proposition 86.13. We have c_{k}(V 𝟙/E) = c_{k}(E^{∨}) = 2e_{k} and c_{k}(E) = (−1)^{k}2e_{k} for all k ∈ [1, n].

Proof. Let s : X → ℙ(V) be the closed embedding. Let H denote the class of a hyperplane in ℙ(V). We have s_{∗}(h^{k}) = 2H^{k+1} for all k ≥ 0.

First suppose that dim φ = 2n + 1. It follows from (86.3) that

[ℙ(E)] = s_{∗}(l_{n−1}) × 1 + ∑_{k=1}^{n} 2H^{n+1−k} × e_{k}

in CH(ℙ(V) × Gr).

On the other hand, by Proposition 58.10 applied to the subbundle E of V 𝟙, we have

[ℙ(E)] = ∑_{k=0}^{n+1} H^{n+1−k} × c_{k}(V 𝟙/E).

It follows from the Projective Bundle Theorem 53.10 that c_{k}(V 𝟙/E) = 2e_{k} for k ∈ [1, n].

By duality, V 𝟙/E^{⊥} ≃ E^{∨}. Note that the line bundle E^{⊥}/E carries a nondegenerate quadratic form, hence is isomorphic to its dual. Since Pic(Gr) = CH^{1}(Gr) is torsion-free, we conclude that E^{⊥}/E ≃ 𝟙. Therefore,

c(E^{∨}) = c(V 𝟙/E^{⊥}) = c(V 𝟙/E).

The proof in the case dim φ = 2n + 2 proceeds along similar lines: one uses the equality (86.4) and the duality isomorphism V 𝟙/E ≃ E^{∨}. □

Remark 86.14. Let φ be a nondegenerate quadratic form that is not necessarily split. By Proposition 86.13, the classes 2e_{k}, k ≥ 1, that are a priori defined over a splitting field of φ, are in fact defined over F.

In order to determine the multiplicative structure of CH(Gr), we present the set of defining relations between the e_{k}. For convenience, we set e_{k} = 0 if k > n.

Since c(V 𝟙/E) ∘ c(E) = c(V 𝟙) = 1 and CH(Gr) is torsion-free, it follows from Proposition 86.13 that

(86.15) e_{k}^{2} − 2e_{k−1}e_{k+1} + 2e_{k−2}e_{k+2} − ⋯ + (−1)^{k−1}2e_{1}e_{2k−1} + (−1)^{k}e_{2k} = 0

for all k ≥ 1.

Proposition 86.16. The equalities (86.15) form a set of defining relations between the generators e_{k} of the ring CH(Gr) over CH^{0}(Gr).

Proof. Let A be the factor ring of the polynomial ring ℤ[t_{1}, t_{2}, …, t_{n}] modulo the ideal generated by polynomials giving the relations (86.15). We claim that the ring homomorphism A → CH(Gr) taking t_{k} to e_{k} is an isomorphism.

Call a monomial t_{1}^{r_{1}}t_{2}^{r_{2}} ⋯ t_{n}^{r_{n}} with r_{i} ≥ 0 basic if r_{k} = 0 or 1 for every k. By Theorem 86.12, it is sufficient to prove that the ring A is generated by classes of basic monomials.

87. THE CHOW RING OF Gr(φ) IN THE GENERAL CASE 361

We define the weight w(m) of a monomial m = t_{1}^{r_{1}}t_{2}^{r_{2}} ⋯ t_{n}^{r_{n}} by the formula

w(m) = ∑_{k=1}^{n} k^{2} · r_{k}

and the weight of a polynomial f(t_{1}, …, t_{n}) over ℤ as the minimum of weights of its nonzero monomials. Clearly, w(m · m′) = w(m) + w(m′). For example, in the formula (86.15), we have w(t_{k}^{2}) = 2k^{2}, w(t_{k−i}t_{k+i}) = 2k^{2} + 2i^{2}, and w(t_{2k}) = 4k^{2}. Thus, t_{k}^{2} is the monomial of the lowest weight in the formula (86.15).

Let f be a polynomial representing an element of the ring A. Applying formula (86.15) to the square of a variable t_{k} in a nonbasic monomial of f of the lowest weight increases the weight but not the degree of f. Since the weight of a polynomial of degree d is at most n^{2}d, we eventually reduce to a polynomial having only basic monomials. □

The relations (86.15) look particularly simple modulo 2:

e_{k}^{2} ≡ e_{2k} mod 2 CH(Gr) for all k ≥ 1.

Proposition 86.17. Let φ be a split nondegenerate quadratic form on V over F of dimension 2n + 2 and let φ′ be a nondegenerate subform of φ on a subspace V′ ⊂ V of codimension 1. Let f denote the morphism Gr(φ) → Gr(φ′) taking U to U ∩ V′, and let e′_{k}, k ≥ 1, denote the standard generators of CH(Gr(φ′)). Then f^{∗}(e′_{k}) = e_{k} for all k ∈ [1, n].

Proof. Denote by E → Gr(φ) and E′ → Gr(φ′) the tautological vector bundles of ranks n + 1 and n respectively. The line bundle

E/f^{∗}(E′) = E/(V′𝟙 ∩ E) ≃ (E + V′𝟙)/V′𝟙 = V 𝟙/V′𝟙

is trivial. In particular, c(E) = c(f^{∗}E′) = f^{∗}c(E′). It follows from Proposition 86.13 that

2f^{∗}(e′_{k}) = (−1)^{k}f^{∗}(c_{k}(E′)) = (−1)^{k}c_{k}(f^{∗}(E′)) = (−1)^{k}c_{k}(E) = 2e_{k}.

The result follows since CH(Gr(φ)) is torsion-free. □

87. The Chow ring of Gr(φ) in the general case

Let φ be a nondegenerate quadratic form of dimension 2n + 1 on V over an arbitrary field F and let X = X_{φ}. Let Y be a smooth proper scheme over F and let h : Y → Gr = Gr(φ) be a morphism. We set E′ = h^{∗}(E), where E is the tautological vector bundle over Gr, and view ℙ(E′) as a closed subscheme of X ×Y.

Proposition 87.1. The CH(Y)-module CH(X × Y) is free with basis h^{k}, h^{k} · [ℙ(E′)] with k ∈ [1, n − 1].

Proof. Let V 𝟙 denote the trivial vector bundle V × Y over Y. We claim that the restriction f : T = (X × Y) \ ℙ(E′) → ℙ(V 𝟙/E′^{⊥}) of the natural morphism f : ℙ(V 𝟙)\ℙ(E′^{⊥}) → ℙ(V 𝟙/E′^{⊥}) is an affine bundle. To do so we use the criterion of Lemma 52.12.

Let R be a local commutative F -algebra. An F -morphism

Spec R → ℙ(V 𝟙/E′^{⊥}),

equivalently, an R-point of ℙ(V 𝟙/E′^{⊥}), determines a pair (U, W) where U is a totally isotropic direct summand of V_{R} of rank n and W is a direct summand of V_{R} of rank n + 2 containing U^{⊥}. Since rank W^{⊥} = n − 1, one can choose an Rbasis of W so that the restriction of the quadratic form φ on W/W^{⊥} is equal to xy + az^{2} for some a ∈ R^{×} and U is given by x = z = 0 in W. Therefore, the fiber Spec R ×_{ℙ(V 𝟙/E′^{⊥})} T is given by the equation y/x = a(z/x)^{2} over R and hence is isomorphic to an affine space over R. By Lemma 52.12, f is an affine bundle.

Thus X × Y is equipped with the structure of a cellular scheme. In particular, we have a (split) exact sequence

0 → CH(ℙ(E′)) ⟶^{i_{∗}} CH(X × Y) → CH(T) → 0

and the isomorphism

f^{∗} : CH(ℙ(V 𝟙/E′^{⊥})) →^{∼} CH(T).

The restriction of the canonical line bundle over ℙ(V) to X ×Y and CH(ℙ(E′)) are also canonical bundles. It follows from the Projective Bundle Theorem 53.10 and the Projection Formula (Proposition 56.9) that the image of i_{∗} is a free CH(Y)- module with basis h^{k} · [ℙ(E′)], k ∈ [0, n − 1].

The geometric description of the canonical line bundle given in §104.C shows that the pull-back with respect to f of the canonical line bundle is the restriction to T of the canonical bundle on X × Y. Again, it follows from the Projective Bundle Theorem 53.10 that CH(T) is a free CH(Y)-module with basis the restrictions of h^{k}, k ∈ [0, n − 1], on T. The statement readily follows. □

Remark 87.2. The proof of Proposition 87.1 gives the motivic decomposition

M(X × Y) = M(ℙ(E′)) ⊕ M(ℙ(V 𝟙/E′^{⊥}))(n).

As in the case of quadrics, we write CH(\overline{Gr}) for the colimit of CH(Gr_{L}) over all field extensions L/F and \overline{CH}(Gr) for the image of CH(Gr) in CH(\overline{Gr}). We say that a cycle α in CH(\overline{Gr}) is rational if it belongs to \overline{CH}(Gr). We use similar notations and definitions for the cycles on Gr^{2}, the classes of cycles modulo 2, etc.

Corollary 87.3. The elements (e_{k} × 1) + (1 × e_{k}) in CH(\overline{Gr} ^{2}) are rational for all k ∈ [1, n].

Proof. Let E_{1} and E_{2} be the two pull-backs of E on Gr^{2} := Gr × Gr. Pulling the formula (86.3) back to \overline{X} × \overline{Gr} ^{2}, we get in CH(\overline{X} × \overline{Gr} ^{2}):

[ℙ(E_{1})] = l_{n−1} × 1 × 1 + ∑_{k=1}^{n} h^{n−k} × e_{k} × 1,

[ℙ(E_{2})] = l_{n−1} × 1 × 1 + ∑_{k=1}^{n} h^{n−k} × 1 × e_{k}.

Therefore the cycle

[ℙ(E_{1})] − [ℙ(E_{2})] = ∑_{k=1}^{n} h^{n−k} × (e_{k} × 1 − 1 × e_{k})

is rational. Applying Proposition 87.1 to the variety Gr^{2}, we have the cycles (e_{k} × 1) − (1 × e_{k}) are also rational. Note that by Proposition 86.13, the cycles 2e_{k} are rational. □

Now consider the Chow groups Ch(Gr), Ch(\overline{Gr}) modulo 2 and write \overline{Ch}(Gr) for the image of Ch(Gr) in Ch(\overline{Gr}). We still write e_{k} for the class of the generator in Ch^{k}(\overline{Gr}).

For every subset I ⊂ [1, n], the rational correspondence

(87.4)  x_{I} = ∏_{k∈I} [(e_{k} × 1) + (1 × e_{k})] ∈ \overline{Ch}(Gr^{2})

defines an endomorphism (x_{I})_{∗} of Ch(\overline{Gr}) taking \overline{Ch}(Gr) into \overline{Ch}(Gr).

Lemma 87.5. For any subsets I, J ⊂ [1, n], we have

(x_{J})_{∗}(e_{I}) = \{ \begin{matrix}e_{I∩J} & if I ∪ J = [1, n], \\ 0 & otherwise, \end{matrix}

in Ch(\overline{Gr}).

Proof. We have x_{J} = ∑ e_{J_{1}} × e_{J_{2}}, where the sum is taken over all subsets J_{1} and J_{2} of [1, n] such that J is the disjoint union of J_{1} and J_{2}. Hence

(x_{J})_{∗}(e_{I}) = ∑ deg(e_{I} · e_{J_{1}}) e_{J_{2}}

and the statement is implied by the following lemma.

Lemma 87.6. For any subsets I, J ⊂ [1, n],

deg(e_{I} · e_{J}) ≡ \{ \begin{matrix}1 mod 2 & if J = [1, n] \ I, \\ 0 mod 2 & otherwise. \end{matrix}

Proof. If J = [1, n] \ I, the product e_{I} · e_{J} = e_{[1, n]} is the class of a rational point of \overline{Gr} by Corollary 86.10, hence deg(e_{I} · e_{J}) = 1. Otherwise modulo 2, e_{I} · e_{J} is either zero or the monomial e_{K} for some K different from [1, n] (one uses the relations between the generators modulo 2). Hence deg(e_{I} · e_{J}) ≡ 0 mod 2. □

The following statement is due to Vishik. We give a proof different from the one in [134].

Theorem 87.7. Let Gr be the variety of maximal isotropic subspaces of a nondegenerate quadratic form of dimension 2n + 1 or 2n + 2. Then the ring \overline{Ch}(Gr) is generated by all e_{k}, k ∈ [0, n], such that e_{k} ∈ \overline{Ch}(Gr).

Proof. By Propositions 85.2 and 86.17, it suffices to consider the case of dimension 2n + 1. It follows from Theorem 86.12 that every element α ∈ \overline{Ch}(Gr) can be written in the form α = ∑ a_{I}e_{I} with a_{I} ∈ ℤ/2ℤ. It suffices to prove the following:

Claim: For every I satisfying a_{I} = 1, we have e_{k} ∈ \overline{Ch}(Gr) for any k ∈ I. To prove the claim, we may assume that α is homogeneous. We establish the

claim by induction on the number of nonzero coefficients of α. Choose I with largest |I| such that a_{I} = 1 and set J = ([1, n] \ I) ∪ {k}. By Lemma 87.5, (x_{J})_{∗}(α) = e_{k} or 1 + e_{k}. Indeed, if a_{I′} = 1 for some I′ ⊂ [1, n] with I′ ∪ J = [1, n], then either I′ = [1, n] \ J and hence (x_{J})_{∗}(e_{I′}) = e_{∅} = 1, or I′ = ([1, n] \ J) ∪ {l} for some l. But since α is homogeneous, we must have l = k. Therefore I′ = I and (x_{J})_{∗}(e_{I′}) = e_{k}.

We have shown that e_{k} ∈ \overline{Ch}(Gr) for all k ∈ I. Therefore, e_{I} ∈ \overline{Ch}(Gr) and α − e_{I} ∈ \overline{Ch}(Gr). By the induction hypothesis, the claim holds for α − e_{I} and therefore for α. □

Exercise 87.8. Prove that the tangent bundle of Gr is canonically isomorphic to Λ^{2}(V 𝟙/E).

88. The invariant J(φ)

In this section, we define a new invariant of nondegenerate quadratic forms. It differs slightly from the one defined in [134].

Let φ be a nondegenerate quadratic form of dimension 2n + 1 or 2n + 2 and set Gr = Gr(φ). As before, let e_{k} be viewed as the class of the generator for Ch(\overline{Gr}). We define a new discrete J-invariant J(φ) as follows:

J(φ) = \{k ∈ [0, n] with e_{k} ∉ \overline{Ch}(Gr)\}.

Recall that e_{0} = 1 if dim φ = 2n + 1, hence in this case J(φ) ⊂ [1, n]. When dim φ = 2n + 2, we have 0 ∈ J(φ) if and only if the discriminant of φ is not trivial.

If dim φ = 2n + 2 and φ′ is a nondegenerate subform of φ of codimension 1, then

J(φ) = \{ \begin{matrix}J(φ′) & if disc φ is trivial, \\ \{0\} ∪ J(φ′) & otherwise. \end{matrix}

For a subset I ⊂ [0, n] let ||I|| denote the sum of all k ∈ I.

Proposition 88.1. The smallest dimension i such that \overline{Ch}_{i}(Gr) ≠ 0 is equal to ||J(φ)||.

Proof. By Theorem 87.7, the product of all e_{k} satisfying k ∉ J(φ) is a nontrivial element of \overline{Ch}(Gr) of the smallest dimension equal to ||J(φ)||. □

Proposition 88.2. A nondegenerate quadratic form φ is split if and only if J(φ) = ∅.

Proof. The “only if” part follows from the definition. Suppose the set J(φ) is empty. Since all the e_{k} are rational, the class of a rational point of \overline{Gr} belongs to \overline{Ch}_{0}(Gr) by Corollary 86.10. It follows that Gr has a closed point of odd degree, i.e., φ is split over an odd degree finite field extension. By Springer’s Theorem (Corollary 18.5), the form φ is split. □

Lemma 88.3. Let φ = φ̃ ⊥ ℍ. Then J(φ) = J(φ̃).

Proof. Suppose that dim φ = 2n + 1. Note first that the cycle e_{n} = [Gr(φ̃)] is rational so that n ∉ J(φ). Let k ≤ n − 1. It follows from the decomposition (86.2) that CH^{k}(Gr) ≃ CH^{k} Gr(φ̃) and e_{k} corresponds to ẽ_{k} by Lemma 86.11. Hence e_{k} ∈ J(φ) if and only if ẽ_{k} ∈ J(φ̃). The case of the even dimension is similar. □

Corollary 88.4. Let φ and φ′ be Witt-equivalent quadratic forms. Then J(φ) = J(φ′).

Lemma 88.5. Let X be a variety, Y a scheme, and n an integer such that the natural homomorphism CH_{i}(X) → CH_{i}(X_{F (y)}) is surjective for every point y ∈ Y and i ≥ dim X − n. Then CH_{j}(Y) → CH_{j}(Y_{F (X)}) is surjective for every j ≥ dim Y − n.

88. THE INVARIANT J(φ) 365

Proof. Using a localization argument similar to that used in the proof of Proposition 52.10, one checks that the top homomorphism in the commutative diagram

CH(X) ⊗ CH(Y) ⟶ CH(X × Y)
↓  ↓
CH(Y)  ⟶ CH(Y_{F (X)})

is surjective in dimensions ≥ dim X + dim Y − n by induction on dim Y. Since the right vertical homomorphism is surjective, so is the bottom homomorphism in dimensions ≥ dim Y − n. □

Let φ be a quadratic form of dimension 2n + 1 or 2n + 2.

Corollary 88.6. The canonical homomorphism CH^{i}(Gr) → CH^{i}(Gr_{F (X)}) is surjective for all i ≤ n − 1.

Proof. Note that X is split over F (y) for every y ∈ Gr. Therefore, CH^{k}(X_{F (y)}) is generated by h^{k} for all k ≤ n − 1 and, hence, the homomorphism CH^{k}(X) → CH^{k}(X_{F (y)}) is surjective. □

Corollary 88.7. J(φ) ∩ [0, n − 1] ⊂ J(φ_{F (X)}) ⊂ J(φ).

The following proposition relates the set J(φ) and the absolute Witt indices of φ. It follows from Corollaries 88.4 and 88.7.

Proposition 88.8. Let φ be a nondegenerate quadratic form of dimension 2n + 1 or 2n + 2. Then

J(φ) ⊂ {n − 𝔧_{0}(φ), n − 𝔧_{1}(φ), …, n − 𝔧_{𝔥(φ)−1}(φ)}.

In particular, |J(φ)| ≤ 𝔥(φ).

Remark 88.9. One can impose further restrictions on J(φ). Choose a nondegenerate form ψ such that one of the forms φ and ψ is a subform of the other of codimension 1 and the dimension of the largest form is even. Then the sets J(φ) and J(ψ) differ by at most one element 0. Therefore, the inclusion in Proposition 88.8 applied to the form ψ gives

J(φ) ⊂ {0, n − 𝔧_{0}(ψ), n − 𝔧_{1}(ψ), …, n − 𝔧_{𝔥(ψ)−1}(ψ)}.

Example 88.10. Suppose that φ is an anisotropic m-fold Pfister form, m ≥ 1. Then J(φ) = {2^{m−1} − 1}. Indeed, 𝔥(φ) = 1, hence J(φ) ⊂ {2^{m−1} − 1} by Proposition 88.8. But J(φ) is not empty by Proposition 88.2.

We write n_{Gr} for the gcd of deg(g) taken over all closed points g ∈ Gr. The ideal n_{Gr} · ℤ is the image of the degree homomorphism CH(Gr) → ℤ. Since φ splits over a field extension of F of degree a power of 2, the number n_{Gr} is a 2-power.

Proposition 88.11. Let φ be a nondegenerate quadratic form of odd dimension. Then

2^{|J(φ)|} · ℤ ⊂ n_{Gr} · ℤ ⊂ ind(C_{0}(φ)) · ℤ.

Proof. For every k ∉ J(φ), let f_{k} be a cycle in \overline{CH}^{k}(Gr) satisfying f_{k} ≡ e_{k} modulo 2 CH^{k}(\overline{Gr}). By Remark 86.14, we have 2e_{k} ∈ \overline{CH}^{k}(Gr) for all k. Let α be the product of all the f_{k} satisfying k ∉ J(φ) and all the 2e_{k} satisfying k ∈ J(φ).

Clearly, α is a cycle in \overline{CH}(Gr) of degree 2^{|J(φ)|}m, where m is an odd integer. The first inclusion now follows from the fact that n_{Gr} is a 2-power.

Let L be the residue field F (g) of a closed point g ∈ Gr. Since φ splits over L, so does the even Clifford algebra C_{0}(φ). It follows that ind C_{0}(φ) divides [L : F] = deg g for all g and therefore divides n_{Gr}. □

Propositions 88.8 and 88.11 yield

Corollary 88.12. Let φ be a nondegenerate quadratic form of dimension 2n + 1. Consider the following statements:

(1) C_{0}(φ) is a division algebra.

(2) n_{Gr} = 2^{n}.

(3) J(φ) = [1, n].

(4) 𝔧_{k} = k for all k = 0, 1, …, n. Then (1) ⇒ (2) ⇒ (3) ⇒ (4).

The following statement is a refinement of the implication (1) ⇒ (3).

Corollary 88.13. Let φ be a nondegenerate quadratic form of odd dimension and ind C_{0}(φ) = 2^{k}. Then [1, k] ⊂ J(φ).

Proof. We proceed by induction on dim φ = 2n + 1. If k = n, i.e., C_{0}(φ) is a division algebra, the statement follows from Corollary 88.12. So we may assume that k < n. Let φ′ be a form over F (φ) Witt-equivalent to φ_{F (φ)} of dimension less than dim φ. The even Clifford algebra C_{0}(φ′) is Brauer-equivalent to C_{0}(φ)_{F (φ)}. Since C_{0}(φ) is not a division algebra, it follows from Corollary 30.10 that ind(C_{0}(φ′)) = ind(C_{0}(φ)) = 2^{k}. By the induction hypothesis, [1, k] ⊂ J(φ′). By Corollaries 88.4 and 88.7, we have J(φ′) = J(φ_{F (φ)}) ⊂ J(φ). □

Exercise 88.14. Let φ be a quadratic form of odd dimension.

(1) Prove that 1 ∈ J(φ) if and only if the even Clifford algebra C_{0}(φ) is not split.

(2) Prove that 2 ∈ J(φ) if and only if ind C_{0}(φ) > 2.

89. Steenrod operations on Ch(Gr(φ))

We calculate the Steenrod operations on Ch(Gr(φ)) as given by Vishik in [134]. Let φ be a nondegenerate quadratic form on V over F of dimension 2n + 1 or 2n + 2, let X be the projective quadric of φ, let Gr be the variety of maximal totally isotropic subspaces of V, and let E be the tautological vector bundle over Gr. Let s : ℙ(E) → Gr and t : ℙ(E) → X be the projections. There is an exact sequence of vector bundles over ℙ(E):

0 → 𝟙 → L_{c} ⊗ s^{∗}(E) → T_{s} → 0,

where L_{c} is the canonical line bundle over ℙ(E) and T_{s} is the relative tangent bundle of s (cf. Example 104.20). Note that L_{c} is the pull-back with respect to t of the canonical line bundle over X, hence c(L_{c}) = 1 + t^{∗}(h), where h ∈ CH^{1}(X) is the class of a hyperplane section of X. By Proposition 86.13, c_{i}(E) is divisible by 2 for all i > 0. It follows that

c(T_{s}) = c(L_{c} ⊗ s^{∗}(E)) ≡ c(L_{c} ⊗ 𝟙^{n}) = c(L_{c})^{n} = (1 + t^{∗}(h))^{n} mod 2.

Theorem 89.1. Let char F ≠ 2 and Gr = Gr(φ) with φ a nondegenerate quadratic form of dimension 2n + 1 or 2n + 2. Then the Steenrod operation Sq_{Gr} : Ch(Gr) → Ch(Gr) of cohomological type satisfies

Sq_{Gr}^{i}(e_{k}) = \binom{k}{i}e_{k+i}

for all i and k ∈ [1, n].

Proof. We have Sq_{X}(l_{n−k}) = (1 + h)^{n+k} · l_{n−k} by Corollary 78.5. It follows from (86.5), Theorem 61.9, and Proposition 61.10 that

Sq_{Gr}(e_{k}) = Sq_{Gr} ∘s_{∗} ∘ t^{∗}(l_{n−k})
= s_{∗} ∘ c(−T_{s}) ∘ Sq_{ℙ(E)} ∘t^{∗}(l_{n−k})
= s_{∗}((1 + t^{∗}h)^{−n} · t^{∗} ∘ Sq_{X}(l_{n−k}))
= s_{∗} ∘ t^{∗}((1 + h)^{−n} · (1 + h)^{n+k} · l_{n−k})
= s_{∗} ∘ t^{∗}((1 + h)^{k} · l_{n−k})
= ∑_{i≥0} \binom{k}{i}s_{∗} ∘ t^{∗}(l_{n−k−i})

= ∑_{i≥0} \binom{k}{i}e_{k+i}.  □

Exercise 89.2. Let φ be an anisotropic quadratic form of even dimension and height 1. Using Steenrod operations, give another proof of the fact that dim(φ) is a 2-power. (Hint: Use Propositions 88.2 and 88.8.)

90. Canonical dimension

Let F be a field and let C be a class of field extensions of F. Call a field E ∈ C generic if for any L ∈ C there is an F -place E ⇀ L (cf. §103).

Example 90.1. Let X be a scheme over F. A field extension L of F is called an isotropy field of X if X(L) ≠ ∅. If X is a smooth variety, it follows from §103 that the field F (X) is generic in the class of all isotropy fields of X.

The canonical dimension cdim(C) of the class C is defined to be the minimum of the tr. deg_{F} E over all generic fields E ∈ C. If X is a scheme over F, we write cdim(X) for cdim(C), where C is the class of fields as defined in Example 90.1. If X is smooth, then cdim(X) ≤ dim X.

Let p be a prime integer and let C be a class of field extensions of F. A field E ∈ C is called p-generic if for any L ∈ C there is an F -place E ⇀ L′ for some finite extension L′ of L of degree prime to p. The canonical p-dimension cdim_{p}(C) of C and cdim_{p}(X) of a scheme X over F are defined similarly. Clearly, cdim_{p}(C) ≤ cdim(C) and cdim_{p}(X) ≤ cdim(X).

The following theorem answers an old question of Knebusch (cf. [83, §4]):

Theorem 90.2. For an arbitrary anisotropic smooth projective quadric X,

cdim_{2}(X) = cdim(X) = dim_{Izh} X.

Proof. Let Y be a smooth subquadric of X of dimension dim Y = dim_{Izh} X. Note that 𝔦_{1}(Y) = 1 by Corollary 74.3. Clearly, the function field F (Y) is an isotropy field of X. Moreover, if L is an isotropy field of X, then by Lemma 74.1, we have Y (L) ≠ ∅. Since the variety Y is smooth, there is an F -place F (Y) ⇀ L (cf. §103). Therefore, F (Y) is a generic isotropy field of X.

Suppose that E is an arbitrary 2-generic isotropy field of X. We show that tr. deg_{F} E ≥ dim Y which will finish the proof.

Since E and F (Y) are both generic isotropy fields of the same X, we have F -places π : F (Y) ⇀ E and ε : E ⇀ E′, where E′ is an odd degree field extension of F (Y). Let y and y′ be the centers of π and ε ∘ π respectively. Clearly, y′ is a specialization of y and therefore,

dim y′ ≤ dim y ≤ tr. deg_{F} E.

The morphism Spec E′ → Y induced by ε ∘ π gives rise to a prime correspondence δ: Y ⇝ Y with odd mult(δ) so that p_{2∗}(δ) = [y′], where p_{2} : Y × Y → Y is the second projection. By Theorem 75.4, mult(δ^{t}) is odd, hence y′ is the generic point of Y and dim y′ = dim Y. □

The rest of this section is dedicated to determining the canonical 2-dimension of the class C of all splitting fields of a nondegenerate quadratic form φ. Note for this class C, we have cdim(C) = cdim(Gr) and cdim_{2}(C) = cdim_{2}(Gr), where Gr = Gr(φ), since L ∈ C if and only if Gr(L) ≠ ∅. Hence

cdim_{2}(Gr) ≤ cdim(Gr) ≤ dim Gr.

Theorem 90.3. Let φ be a nondegenerate quadratic form over F. Then

cdim_{2}(Gr(φ)) = ||J(φ)||.

Proof. Let E be a 2-generic isotropy field of Gr such that tr. deg_{F} E = cdim_{2}(Gr). As E is an isotropy field, there is a morphism Spec E → Gr over F. Let Y be the closure of the image of this morphism. We view F (Y) as a subfield of E. Clearly, tr. deg_{F} E ≥ dim Y.

Since E is 2-generic, there is a field extension L/F (Gr) of odd degree and an F -place E ⇀ L. Restricting this place to the subfield F (Y), we get a morphism f : Spec L → Y as Y is complete. Let g : Spec L → Gr be the morphism induced by the field extension L/F (Gr). Then the closure Z of the image of the morphism (f, g) : Spec L → Y × Gr is of odd degree [L : F (Gr)] when projecting to Gr. Therefore, the image of [Z] under the composition

Ch(Y × Gr) ⟶^{(i×1_{Gr})_{∗}} Ch(Gr × Gr) ⟶^{q_{∗}} Ch(Gr),

where i : Y → Gr is the closed embedding and q is the second projection, is equal to [Gr]. In particular, (i × 1_{Gr})_{∗}([Z]) ≠ 0, hence (i × 1_{Gr})_{∗} ≠ 0.

We claim that the push-forward homomorphism i_{∗} : Ch(Y) → Ch(Gr) is also nontrivial. Let L be the residue field of a point of Y. Consider the induced morphism j : Spec L → Gr. The pull-back of the element x_{I} in \overline{Ch}(Gr^{2}) (defined in (87.4)) with respect to the morphism j × 1_{Gr} : Gr_{L} → Gr^{2} is equal to e_{I} ∈ \overline{Ch}(Gr_{L}) = Ch(Gr_{L}). Since the elements e_{I} generate Ch(Gr_{L}) by Theorem 86.12, the pull-back homomorphism Ch(Gr^{2}) → Ch(Gr_{L}) is surjective. Applying Proposition 58.18 to the projection p : Y × Gr → Y and the embedding i × 1_{Gr} : Y × Gr → Gr^{2}, we conclude that the product

h_{Y} : Ch(Y) ⊗ Ch(Gr^{2}) → Ch(Y × Gr), α ⊗ β ↦ p^{∗}(α) · β

is surjective.

By Propositions 49.20 and 58.17, the diagram

Ch(Y) ⊗ Ch(Gr^{2}) ⟶^{h_{Y}} Ch(Y × Gr)
i_{∗}⊗1↓  ↓^{(i×1_{Gr})_{∗}}
Ch(Gr) ⊗ Ch(Gr^{2}) ⟶^{h_{Gr}} Ch(Gr × Gr)

is commutative. As (i × 1_{Gr})_{∗} is nontrivial, we conclude that i_{∗} is also nontrivial. This proves the claim.

By Proposition 88.1, we have dim Y ≥ ||J(φ)||, hence

cdim_{2}(Gr) = tr. deg_{F} E ≥ dim Y ≥ ||J(φ)||.

It follows from Proposition 88.1 that there is a closed subvariety Y ⊂ Gr of dimension ||J(φ)|| such that [Y] ≠ 0 in \overline{Ch}(Gr) ⊂ Ch(Gr_{F (Gr)}). By Lemma 87.6, there is a β ∈ Ch(Gr_{F (Gr)}) satisfying [Y] · β ≠ 0 in Ch_{0}(Gr_{F (Gr)}). It follows from Proposition 56.11 that the product [Y] · β belongs to the image of the push-forward homomorphism

Ch_{0}(Y_{F (Gr)}) → Ch_{0}(Gr_{F (Gr)}),

therefore Ch_{0}(Y_{F (Gr)}) ≠ 0. In other words, there is a closed point y ∈ Y_{F (Gr)} of odd degree. Let Z be the closure of the image of y under the canonical morphism Y_{F (Gr)} → Y × Gr. Note that that the projection Z → Gr is of odd degree deg(y), hence F (Z) is an extension of F (Gr) of odd degree. Let Y′ denote the image of the projection Z → Y, so F (Y′) is isomorphic to a subfield of F (Z).

We claim that F (Y′) is a 2-generic splitting field of Gr. Indeed, since Y′ is a subvariety of Gr, the field F (Y′) is a splitting field of Gr. Let L be another splitting field of Gr. By Lemma 103.2, a geometric F -place F (Gr) ⇀ L can be extended to an F -place F (Z) ⇀ L′ where L′ is an extension of L of odd degree. Restricting to F (Y′), we get an F -place F (Y′) ⇀ L′. This proves the claim. Therefore, we have

cdim_{2}(Gr) ≤ dim Y′ ≤ dim Y = ||J(φ)||.  □

Theorem 90.3 and Corollary 88.12 yield

Corollary 90.4 (cf. [74, Th. 1.1]). Let φ be a nondegenerate quadratic form of dimension 2n + 1 such that J(φ) = [1, n] (e.g., if C_{0}(φ) is a division algebra or if n_{Gr} = 2^{n}). Then

cdim_{2}(Gr) = cdim(Gr) = dim(Gr) = \frac{n(n + 1)}{2}.

Example 90.5. Let φ be an anisotropic m-fold Pfister form with m ≥ 1. Since the class of splitting fields of φ coincides with the class of isotropy fields, we have cdim(Gr) = dim_{Izh}(X) = 2^{m−1} − 1. By Theorem 90.3 and Example 88.10, we have cdim_{2}(Gr) = ||J(φ)|| = 2^{m−1} − 1.

We next compute the canonical dimensions cdim(Gr), cdim_{2}(Gr) and determine the set J(φ) for an excellent quadratic form φ. Write the dimension of φ in the form

(90.6)  dim φ = 2^{p_{0}} − 2^{p_{1}} + 2^{p_{2}} − ⋯ + (−1)^{s−1}2^{p_{s−1}} + (−1)^{s}2^{p_{s}}

with some integers p_{0}, p_{1}, …, p_{s} satisfying p_{0} > p_{1} > ⋯ > p_{s−1} > p_{s} + 1 > 0. Note that the height 𝔥 of φ equals s + 1 for even dim φ, while 𝔥 = s if dim φ is odd.

Let ψ be the p_{𝔥−1}-fold Pfister form ρ_{𝔥−1} over F defined in Theorem 28.3 for the form φ. Since φ and ψ have the same classes of splitting fields, we have cdim Gr(φ) = cdim Gr(ψ) and cdim_{2} Gr(φ) = cdim_{2} Gr(ψ). By Example 90.5,

(90.7)  cdim(Gr(φ)) = cdim_{2}(Gr(φ)) = 2^{p_{𝔥−1}−1} − 1.

Proposition 90.8. Let φ be an anisotropic excellent form of height 𝔥. Then J(φ) = {2^{p_{𝔥−1}−1} − 1}, where the integer p_{𝔥−1} is determined in (90.6).

Proof. Note that j_{𝔥−1} = (dim φ − dim ψ)/2 if dim φ is even and j_{𝔥−1} = (dim φ − dim ψ + 1)/2 if dim φ is odd. Hence by Proposition 88.8, every element of J(φ) is at least 2^{p_{𝔥−1}−1} − 1. By Theorem 90.3, we have cdim_{2}(Gr(φ)) = ||J(φ)||. It follows from (90.7) that J(φ) = {2^{p_{𝔥−1}−1} − 1}. □

The notion of canonical dimension of algebraic groups was introduced by Berhuy and Reichstein in [17]. Here we have presented the more general notion of canonical dimension and p-canonical dimension of a class of field extensions of a given field (cf. [77]). The computation of cdim_{2}(Gr(φ)) given in Theorem 90.3 is new. It is conjectured in [134, Conj. 6.6] that cdim(Gr(φ)) = cdim_{2}(Gr(φ)).

CHAPTER XVII Motives of Quadrics

91. Comparison of some discrete invariants of quadratic forms

In this section, F is an arbitrary field, n a positive integer, V a vector space over F of dimension 2n or 2n + 1, φ a nondegenerate quadratic form on V, X the projective quadric of φ. For any positive integer i, we write G_{i} for the scheme of i-dimensional totally isotropic subspaces of V. In particular, G_{1} = X and G_{i} = ∅ for i > n.

We write Ch(Y) for the Chow group modulo 2 of an F -scheme Y; Ch(Ȳ) is the colimit of Ch(Y_{L}) over all field extensions L/F, \overline{Ch}(Y) is the reduced Chow group, i.e., the image of the homomorphism Ch(Y) → Ch(Ȳ).

We write \overline{Ch}(G_{∗}) for the direct sum ∐_{i≥1} \overline{Ch}(G_{i}). We recall that \overline{Ch}(X^{∗}) stands for ∐_{i≥1} \overline{Ch}(X^{i}), where X^{i} is the direct product of i copies of X. We consider \overline{Ch}(G_{∗}) and \overline{Ch}(X^{∗}) as invariants of the quadratic form φ. Note that their components \overline{Ch}(G_{i}) and \overline{Ch}(X^{i}) are subsets of the finite sets Ch(Ḡ_{i}) and Ch(X̄^{i}) respectively, depending only on dim φ.

These invariants are not independent. A relation between them is described in the following theorem:

Theorem 91.1. The following three invariants of a nondegenerate quadratic form φ of a fixed dimension are equivalent in the sense that if φ′ is another nondegenerate quadratic form with dim φ = dim φ′ and the values of one of the invariants for φ and φ′ are equal, then the values of any other of the invariants for φ and φ′ are also equal.

(i) \overline{Ch}(X^{∗}),

(ii) \overline{Ch}(X^{n}),
(iii) \overline{Ch}(G_{∗}).

Remark 91.2. Although the equivalence of the above invariants means that any of them can be expressed in terms of any other, it does not seem to be possible to get manageable formulas relating (iii) with (ii) or (i).

We need some preparation to prove Theorem 91.1. For i ≥ 1, write Fl_{i} for the scheme of flags V_{1} ⊂ ⋯ ⊂ V_{i} of totally isotropic subspaces V_{1}, …, V_{i} of V, where dim V_{j} = j. In particular, Fl_{1} = X and Fl_{i} = ∅ for i > n. The following lemma generalizes Example 66.6:

Lemma 91.3. For any i ≥ 1, the product Fl_{i} ×X has the canonical structure of a relative cellular scheme where the bases of the cells are as follows: a rank i − 1 projective bundle over Fl_{i}, the scheme Fl_{i+1}, and the scheme Fl_{i} taken i times.

Proof. We construct a descending cellular filtration

Y = Y^{(0)} ⊃ Y^{(1)} ⊃ ⋯ ⊃ Y^{(i+1)} ⊃ Y^{(i+2)} = ∅

on the scheme Y = Fl_{i} ×X as follows: for j ∈ [1, i] the scheme Y^{(j)} is the subscheme of pairs

(91.4)  (V_{1} ⊂ ⋯ ⊂ V_{i}, W)

such that the subspace W + V_{j} is totally isotropic and Y^{(i+1)} is the subscheme of the pairs (91.4) with W ⊂ V_{i}.

The natural projection Y^{(j−1)} \Y^{(j)} → Fl_{i} is a vector bundle for j ∈ [1, i]. The map Y^{(i)} \ Y^{(i+1)} → Fl_{i+1} taking a pair (91.4) to the flag V_{1} ⊂ ⋯ ⊂ V_{i} ⊂ V_{i} + W is also a vector bundle. The projection of the scheme Y^{(i+1)} onto Fl_{i} is a (rank i − 1) projective bundle. Of course, if i ≥ n, then Y^{(i+1)} = Y^{(i)} (and the base of the empty “cell” Y^{(i)} \ Y^{(i+1)} is the empty scheme Fl_{i+1}). □

Corollary 91.5. The motive of the product Fl_{i} ×X for i ≤ n canonically decomposes as a direct sum, where each summand is some shift of the motive of the scheme Fl_{i} or of the scheme Fl_{i+1}. Moreover, a shift of the motive of Fl_{i} occurs in the decomposition and a shift of the motive of Fl_{i+1} also occurs (if i + 1 ≤ n).

Proof. By Corollary 66.4 and Lemma 91.3, the motive of Fl_{i} ×X decomposes into a direct sum of summands that are shifts of the motives of Y^{(i+1)}, Fl_{i+1}, and Fl_{i}, where Y^{(i+1)} is a projective bundle over Fl_{i}. By Theorem 63.10, the motive of Y^{(i+1)} is also a direct sum of shifts of the motive of Fl_{i}. □

Corollary 91.6. For any r ≥ 1, the motive of X^{r} canonically decomposes into a direct sum, where each summand is a shift of the motive of some Fl_{i} with i ∈ [1, r]. Moreover, for any i ∈ [1, r] with i ≤ n, a shift of the motive of Fl_{i} occurs.

Proof. We induct on r. Since X^{1} = X = Fl_{1}, the case r = 1 is immediate. If the statement is proved for some r ≥ 1, then the statement for X^{r+1} follows by Corollary 91.5. □

Lemma 91.7. For any i ≥ 1, the motive of Fl_{i} canonically decomposes into a direct sum, where each summand is a shift of the motive of the scheme G_{i}.

Proof. For each j ∈ [1, i], write Φ_{j} for the scheme of flags V_{1} ⊂ ⋯ ⊂ V_{i−j} ⊂ V_{i} of totally isotropic subspaces V_{k} of V satisfying dim V_{k} = k for any k. In particular, Φ_{1} = Fl_{i} and Φ_{i} = G_{i}. The projections

Fl_{i} = Φ_{1} → Φ_{2} → ⋯ → Φ_{i} = G_{i}

are projective bundles. Therefore, the lemma follows from Theorem 63.10. □

Combining Corollary 91.6 with Lemma 91.7, we get

Corollary 91.8. For any r ≥ 1, the motive of X^{r} canonically decomposes into a direct sum, where each summand is a shift of the motive of some G_{i} with i ∈ [1, r]. Moreover, for any i ∈ [1, r] with i ≤ n, a shift of the motive of G_{i} occurs.

Proof of Theorem 91.1. The equivalences (i) ⇔ (iii) and (ii) ⇔ (iii) are given by Corollary 91.8. □

Remark 91.9. One may say that the invariant \overline{Ch}(X^{n}) is a “compact form” of the invariant \overline{Ch}(X^{∗}) and also that the invariant \overline{Ch}(G_{∗}) is a “compact form” of \overline{Ch}(X^{n}). However, some properties of these invariants are easier formulated and proven on the level of \overline{Ch}(X^{∗}). Among such properties (used many times above), we have the stability of \overline{Ch}(X^{∗}) ⊂ Ch(X̄^{∗}) under partial operations on Ch(X̄^{∗}) given by permutations of factors of any X^{r} as well as pull-backs and push-forwards with respect to partial projections and partial diagonals between X^{r} and X^{r+1}. It is also easier to describe a basis of Ch(X̄^{∗}) and compute multiplication and Steenrod operations (giving further restrictions on \overline{Ch}(X^{∗})) in terms of this basis, than to do a similar analysis for Ch(Ḡ_{∗}).

92. The Nilpotence Theorem for quadrics

Let Λ be a commutative ring. We shall work in the categories CR_{∗}(F, Λ) and CR(F, Λ), introduced in §63.

Let C be a class of smooth complete schemes over field extensions of F closed under taking finite disjoint unions (of schemes over the same field), taking connected components, and scalar extensions. We say that C is tractable if for any variety X in C having a rational point and of positive dimension, there is a scheme X′ in C satisfying dim X′ < dim X and M(X′) ≃ M(X) in CR_{∗}(F, Λ). A scheme is called tractable, if it is member of a tractable class.

Our primary example of a tractable scheme will be any smooth projective quadric over F, the tractable class being the class of (all finite disjoint unions of) all smooth projective quadrics over field extensions of F (cf. Example 66.7).

A smooth projective scheme is called split if its motive in CR_{∗}(F, Λ) is isomorphic to a finite direct sum of several copies of the motive Λ. Any tractable scheme X splits over an extension of the base field. Moreover, the number of copies of Λ in the corresponding decomposition is an invariant of X. We call this the rank of X and denote it by rk X. The number of components of any tractable scheme does not exceed its rank.

Exercise 92.1. Let X/F be a smooth complete variety such that for any field extension E/F satisfying X(E) ≠ ∅, the scheme X_{E} is split (for example, the variety of the maximal totally isotropic subspace of a nondegenerate odd-dimensional quadratic form considered in Chapter XVI). Show that X is tractable.

Exercise 92.2. Show that the product of two tractable schemes is tractable.

Remark 92.3. As shown in [25], the class of all projective homogeneous varieties (under the action of an algebraic group) is tractable.

The following theorem was initially proved by Rost in the case of quadrics (cf. Theorem 67.1). The more general case of a projective homogeneous variety was done in [25].

Theorem 92.4 (Nilpotence Theorem for tractable schemes). Let X be a tractable scheme over F with M(X) its motive in CR_{∗}(F, Λ) or in CR(F, Λ) and α ∈ End M(X) a correspondence. If α_{E} ∈ End M(X_{E}) vanishes for some field extension E/F, then α is nilpotent.

Proof. It suffices to consider the case of the category CR_{∗}(F, Λ) as the functor (63.3) is faithful. We fix a tractable class of schemes containing X. We shall construct a map

N : [0, +∞) × [1, rk X] → [1, +∞)

with the following properties. If Y is a scheme in the tractable class with rk Y ≤ rk X and α ∈ CH(Y^{2}; Λ) a correspondence vanishing over some field extension of F, then α^{N(i,j)} = 0 if dim Y ≤ i and the number of i-dimensional connected components of Y is at most j.

If dim Y = 0, then any extension of scalars induces an injection of CH(Y^{2}; Λ). In this case, we set N(0, j) = 1 for any j ≥ 1.

Now order the set [0, +∞) × [1, rk X] lexicographically. Let (i, j) be a pair with i ≥ 1. Assume that N has been defined on all pairs smaller than (i, j).

Let Y be an arbitrary scheme in the class such that dim Y = i and the number of the i-dimensional components of Y is j. To simplify the notation, we shall assume that the field of definition of Y is F. Let Y_{1} be a fixed i-dimensional component of Y and set Y_{0} to be the union of the remaining components of Y. We take an arbitrary correspondence α ∈ CH(Y^{2}; Λ) vanishing over a scalar extension and replace it by α^{N(i′,j′)}, where (i′, j′) is the pair preceding (i, j) in the lexicographical order. Then for any point y ∈ Y_{1}, we have α_{F (y)} = 0, since the motive of the scheme Y_{F (y)} is isomorphic to the motive of another scheme having j − 1 components of dimension i. Applying Theorem 67.1, we see that

α^{i+1} ∘ CH(Y_{1} × Y; Λ) = 0.

In particular, the composite of the inclusion morphism M(Y_{1}) → M(Y) with α^{i+1} is trivial. Replace α by α^{i+1}. We can view α as a 2 × 2 matrix according to the decomposition M(Y) ≃ M(Y_{0}) ⊕ M(Y_{1}). Its entries corresponding to Hom(M(Y_{1}), M(Y_{0})) and to End M(Y_{1}) are 0. Moreover, the matrix entry corresponding to End M(Y_{0}) is nilpotent with N(i′, j′) its nilpotence exponent, as the number of the i-dimensional components of Y_{0} is at most j − 1. Replacing α by α^{N(i′,j′)} , we may assume that α has only one possibly nonzero entry, namely, the (nondiagonal) entry corresponding to Hom(M(Y_{0}), M(Y_{1})). Therefore, α^{2} = 0. Set N(i, j) = 2(i + 1)N(i′, j′)^{2}. We have shown that for any scheme Y in the tractable class with rk Y ≤ rk X and any correspondence α ∈ CH(Y^{2}; Λ) vanishing over some field extension of F we have α^{N(i,j)} = 0 if dim Y = i and the number of idimensional connected components of Y is j. Since N(i, j) ≥ N(i′, j′), one also has α^{N(i,j)} = 0 if dim Y ≤ i and the number of i-dimensional connected components of Y is smaller than j. □

Corollary 92.5. Let X be a tractable scheme over F and let E/F be a field extension. If q ∈ End M(X_{E}) is an idempotent lying in the image of the restriction End M(X) → End M(X_{E}) in the motivic category CR_{∗}(F, Λ) or CR(F, Λ), then there exists an idempotent p ∈ End M(X) satisfying p_{E} = q.

Proof. Choose a correspondence p′ ∈ End M(X) satisfying p′_{E} = q. Let A (respectively, B) be the (commutative) subring of

End M(X) (respectively, End(M(X_{E})))

generated by p′ (respectively, q). By Theorem 92.4, the kernel of the ring epimorphism A ↠ B consists of nilpotent elements. It follows that the map Spec B → Spec A is a homeomorphism and, in particular, induces a bijection of the sets of the connected components of these topological spaces. Therefore, the homomorphism A → B induces a bijection of the sets of idempotents of these rings (cf. [18, Ch. II, §4.3, Prop. 15, Cor. 1]), and we can find a required p inside of A. □

Exercise 92.6. Show that one can take for p some power of p′. (Hint: Prove and use the fact that the kernel of End X → End X_{E} is annihilated by a positive integer.)

Corollary 92.7. Let X and Y be tractable schemes and let p ∈ End M(X) and q ∈ End M(Y) be idempotents in the motivic category CR_{∗}(F, Λ) or CR(F, Λ). Let f be a morphism (X, p) → (Y, q) in the category CM_{∗}(F, Λ) or CM(F, Λ) respectively. If f_{E} is an isomorphism for some field extension E/F, then f is also an isomorphism.

Proof. By Proposition 63.4, it suffices to prove the result for the category CR_{∗}(F, Λ).

First suppose that Y = X and q = p. We may assume that the scheme X_{E} is split and we have fixed an isomorphism of the motive (X_{E}, p_{E}) with the direct sum of n copies of Λ for some n. Then Aut(X_{E}, p_{E}) = GL_{n}(Λ). Let P (t) ∈ Λ[t] be the characteristic polynomial of the matrix f_{E}, hence P (f_{E}) = 0. If Q(t) ∈ Λ[t] satisfies P (t) = P (0) + tQ(t), the endomorphism

f_{E} ∘ Q(f_{E}) = Q(f_{E}) ∘ f_{E} = P (f_{E}) − P (0) = −P (0) = ± det f_{E}

is multiplication by an invertible element ε = ± det f_{E} in the coefficient ring Λ. By Theorem 92.4, the endomorphism α ∈ End(X, p) satisfying f ∘ Q(f) = Q(f) ∘ f = ε + α is nilpotent. Thus the composites f ∘ Q(f) and Q(f) ∘ f are automorphisms. Consequently f is an automorphism.

In the general case, consider the transpose f^{t} : (Y, q) → (X, p) of f. Since f_{E} is an isomorphism, f_{E}^{t} is also an isomorphism. It follows by the previous case that the composites f ∘f^{t} and f^{t} ∘f are automorphisms. Thus f is an isomorphism. □

Corollary 92.8. Let X be a tractable scheme with p, p′ ∈ End M(X) idempotents satisfying p_{E} = p′_{E} for some field E ⊃ F. Then the motives (X, p) and (X, p′) are canonically isomorphic.

Proof. The morphism p′ ∘ p : (X, p) → (X, p′) is an isomorphism because it becomes an isomorphism over E. □

93. Criterion of isomorphism

In this section, we let Λ = ℤ/2ℤ.

Theorem 93.1. Let X and Y be smooth projective quadrics over F. Then the motives of X and Y in the category CR(F, ℤ/2ℤ) are isomorphic if and only if dim X = dim Y and 𝔦_{0}(X_{L}) = 𝔦_{0}(Y_{L}) for any field extension L/F.

Proof (cf. [71]). The “only if” part of the statement is easy: the motive M(X) of X in CR(F, ℤ/2ℤ) determines the graded group Ch^{∗}(X) which in turn determines dim X and 𝔦_{0}(X) by Corollary 72.6. To prove the “if” part assume that dim X = dim Y and 𝔦_{0}(X_{L}) = 𝔦_{0}(Y_{L}) for any field extension L/F. As before we write D for dim X and set d = [D/2].

The case of split X and Y is trivial. Note that in the split case an isomorphism M(X) → M(Y) is given by the cycle c_{XY} + deg(l_{d}^{2})(h^{d} × h^{d}), where

c_{XY} = ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}) ∈ Ch(X × Y)

(cf. Lemma 73.1). By Corollary 92.7, it follows in the nonsplit case that the motives of X and Y are isomorphic if the cycle c_{XY} ∈ Ch(X̄ × Ȳ) is rational.

To prove Theorem 93.1 in the general case, we show by induction on D that the cycle c_{XY} is rational.

If X (and therefore Y) is isotropic, then the cycle c_{X_{0}Y_{0}} is rational by the induction hypothesis, where X_{0} and Y_{0} are the anisotropic parts of X and Y respectively. It follows that the cycle c_{XY} is rational in the isotropic case. We may therefore assume that X and Y are anisotropic.

To finish the proof we need two results. For their proofs we introduce some special notation and terminology. Write N for the set of the symbols

{h^{i} × l_{i}, l_{i} × h^{i}}_{i∈[0, d]}.

For any subset I ⊂ N, write c_{XY} (I) for the sum of the basis elements of Ch^{D}(X̄ × Ȳ) corresponding to the symbols of I. Similarly, define the cycles c_{Y X}(I) ∈ Ch^{D}(Ȳ × X̄), c_{XX}(I) ∈ Ch^{D}(X̄^{2}), and c_{Y Y} (I) ∈ Ch^{D}(Ȳ^{2}).

We call a subset I ⊂ N admissible, if the cycles c_{XY} (I) and c_{Y X}(I) are rational and weakly admissible if c_{XX}(I) and c_{Y Y} (I) are rational.

Since the set N is weakly admissible by Lemma 73.1, the complement N \ I of any weakly admissible set I is also weakly admissible.

Call a subset I ⊂ N symmetric, if it is stable under transposition, i.e., I^{t} = I. For any I ⊂ N, the set I ∪ I^{t} is the smallest symmetric set containing I; we call it the symmetrization of I.

Lemma 93.2. (1) Any admissible set is weakly admissible.

(2) The symmetrization of an admissible set is admissible.

(3) A union of admissible sets is admissible.

Proof. (1): This follows from the formulas which hold up to the addition of h^{d} × h^{d}:

c_{XX}(I) = c_{Y X}(I) ∘ c_{XY} (I) and c_{Y Y} (I) = c_{XY} (I) ∘ c_{Y X}(I).

(3): Let I and J be admissible sets. The cycle c_{XY} (I ∪ J) is rational as

c_{XY} (I ∪ J) = c_{XY} (I) + c_{XY} (J) + c_{XY} (I ∩ J)

and up to the addition of h^{d} × h^{d}, we have c_{XY} (I ∩ J) = c_{XY} (J) ∘ c_{XX}(I). The rationality of c_{Y X}(I ∪ J) is proved analogously.

(2): The transpose I^{t} of an admissible set I ⊂ N is admissible. Therefore, by (3), the union I ∪ I^{t} is admissible. □

The key observation is:

Proposition 93.3. Let I be a weakly admissible set and let h^{r} × l_{r} ∈ I be the element with smallest r. Then h^{r} × l_{r} is contained in an admissible set.

Assuming Proposition 93.3, we finish the proof of Theorem 93.1 by showing that the set N is admissible.

Note that ∅ is a symmetric admissible set. Let I_{0} be a symmetric admissible set. It suffices to show that if I_{0} ≠ N, then I_{0} is contained in a strictly bigger symmetric admissible set I_{1}.

By Lemma 93.2(1), the set I_{0} is weakly admissible. Therefore, the set I := N\I_{0} is weakly admissible as well. Since the set I is nonempty and symmetric, h^{i} ×l_{i} ∈ I for some i. Take the smallest r with h^{r} × l_{r} ∈ I. By Proposition 93.3, there is an admissible set J containing h^{r} × l_{r}. By Lemma 93.2(3), the union I_{0} ∪ J is also an admissible set. Let I_{1} be its symmetrization. Then the set I_{1} is admissible by Lemma 93.2(2). It is symmetric and contains I_{0} properly because h^{r} × l_{r} ∈ I_{1} \ I_{0}.

So to finish, we must only prove Proposition 93.3.

Proof of Proposition 93.3. Multiplying the generic point morphism

Spec F (X) → X

by X × Y (on the left), we get a flat morphism

(X × Y)_{F (X)} → X × Y × X.

This induces a surjective pull-back homomorphism

f : Ch^{D}(X̄ × Ȳ × X̄) → Ch^{D}(X̄ × Ȳ)

mapping each basis element of the form β_{1} × β_{2} × h^{0} to β_{1} × β_{2} and vanishing on the remaining basis elements. Note that this homomorphism maps the subgroup of F -rational cycles onto the subgroup of F (X)-rational cycles (cf. Corollary 57.11).

Since the quadrics X_{F (X)} and Y_{F (X)} are isotropic, the cycle c_{XY} (N) is F (X)- rational. Therefore, the set f^{−1}(c_{XY} (N)) contains a rational cycle. Any cycle in this set has the form

(93.4)  c := c_{XY} (N) × h^{0} + ∑ α × β × γ,

where the sum is taken over some homogeneous cycles α, β, γ with codim γ positive. In the following, we assume that (93.4) is a rational cycle.

Let I and r be as in the statement of Proposition 93.3. Viewing the cycle (93.4) as a correspondence from X̄ to Ȳ × X̄, we may take the composition c ∘ c_{XX}(I). The result is a rational cycle on X̄ × Ȳ × X̄ that (up to the addition of h^{d} × h^{d}) is equal to

(93.5)  c′ := c_{XY} (I) × h^{0} + ∑ α × β × γ,

where the sum is taken over some (other) homogeneous cycles α, β, γ with codim γ > 0 and codim α ≥ r. Take the pull-back of the cycle c′ with respect to the morphism X̄ × Ȳ → X̄ × Ȳ × X̄, given by (x, y) ↦ (x, y, x), that is induced by the diagonal of X̄. The result is a rational cycle on X̄ × Ȳ that is equal to

(93.6)  c′′ := c_{XY} (I) + ∑(α · γ) × β,

where codim(α · γ) > r. It follows that c′′ = c_{XY} (J′) for some set J′ containing h^{r} × l_{r}.

Repeating the procedure with X and Y interchanged, we can find a set J′′ containing h^{r} × l_{r} with the cycle c_{Y X}(J′′) rational. Then the set J := J′ ∩ J′′ contains h^{r} × l_{r} and is admissible as c_{XY} (J) coincides (up to addition of h^{d} × h^{d}) with the composition c_{XY} (J′) ∘ c_{Y X}(J′′) ∘ c_{XY} (J′) and a similar equality holds for c_{Y X}(J). □

The proof of Theorem 93.1 is now complete. □

Remark 93.7. By Theorem 27.3, an isomorphism of motives of odd-dimensional quadrics gives rise to an isomorphism of the corresponding varieties. The question whether for a given even n the conditions

n = dim φ = dim ψ and 𝔦_{0}(φ_{L}) = 𝔦_{0}(ψ_{L}) for any L

imply that φ and ψ are similar is answered positively if characteristic is not 2 and n ≤ 6 in [62], and negatively for all n ≥ 8 but 12 in [63]. It remains open for n = 12.

94. Indecomposable summands

In this section, Λ = ℤ/2ℤ and we work in the category CM(F, ℤ/2ℤ) of graded motives. Let X be a smooth anisotropic projective quadric of dimension D. We write P for the set of idempotents in Ch_{D}(X^{2}) = End M(X). We shall provide some information about objects (X, p) (with p ∈ P) in the category CM(F, ℤ/2ℤ). For such p (or, more generally, for any element p ∈ Ch_{D}(X^{2})), let p̄ stand for the essence (as defined in §72) of the image of p in the reduced Chow group \overline{Ch}(X^{2}). We write [(X, p)] for the isomorphism class of the motive (X, p).

Theorem 94.1. (1) The map

\{[(X, p)]\}_{p∈P} → \overline{Ch}_{D}(X^{2}) given by [(X, p)] ↦ p̄

is well-defined and injective; its image is the group \overline{Ch}e_{D}(X^{2}) of all D-dimensional essential cycles.

(2) Let p, p_{1}, p_{2} ∈ P. Then (X, p) ≃ (X, p_{1}) ⊕(X, p_{2}) if and only if p̄ is a disjoint union of p̄_{1} and p̄_{2}, i.e., p̄_{1} and p̄_{2} do not intersect and p̄ = p̄_{1} + p̄_{2}. In particular, the motive (X, p) is indecomposable if and only if the cycle p̄ is minimal.

(3) For any p, p′ ∈ P, the motives (X, p) and (X, p′) are isomorphic to twists of each other if and only if p̄ and p̄′ are derivatives of the same rational cycle. More precisely, if i ≥ 0, then (X, p) ≃ (X, p′)(i) if and only if p̄ = (h^{0} × h^{i}) · α and p̄′ = (h^{i} × h^{0}) · α for some α ∈ \overline{Ch}_{D+i}(X^{2}).

Proof. Let E/F be a field extension such that the quadric X_{E} is split.

(1): By Corollary 92.7, we have [(X, p)] = [(X, p′)] if and only if [(X, p)_{E}] = [(X, p′)_{E}]. Assume that D is odd. Then the basis of Ch(X_{E}^{2}) is a system of orthogonal idempotents. Therefore any element α ∈ Ch(X_{E}^{2}) is an idempotent and the motive (X_{E}, α) is isomorphic to the direct sum of the motives (X_{E}, β) over all basis elements β appearing in the decomposition of α. Besides, (X_{E}, h^{i} × l_{i}) ≃ ℤ/2ℤ(i) and (X_{E}, l_{i} × h^{i}) ≃ ℤ/2ℤ(D − i) for any i ∈ [0, d]. It follows that the map in (1) is well-defined and injective. The statement on the image of the map is a consequence of Corollary 92.5.

Now assume that D is even. In this case the basis elements

{h^{i} × l_{i}, l_{i} × h^{i}}_{i∈[0, d−1]}

of Ch(X_{E}^{2}) are orthogonal idempotents, (X_{E}, h^{i} ×l_{i}) ≃ ℤ/2ℤ(i) and (X_{E}, l_{i} ×h^{i}) ≃ ℤ/2ℤ(D−i) for any i ∈ [0, d−1]. Moreover, each of these idempotents is orthogonal to each of the remaining basis elements h^{d} × l_{d}, l_{d} × h^{d}, h^{d} × h^{d}, and l_{d} × l_{d}. The elements h^{d} ×l_{d}, h^{d} ×l_{d} +h^{d} ×h^{d}, l_{d} ×h^{d} and l_{d} ×h^{d} +h^{d} ×h^{d} are also idempotents, each of the four motives produced by these idempotents is isomorphic to ℤ/2ℤ(d).

If the even integer D is not divisible by 4, then l_{d}^{2} = 0 (cf. Exercise 68.3) and it follows that the idempotents h^{d} ×l_{d} and l_{d} ×h^{d} are orthogonal and their sum is the fifth (the last) nonzero idempotent in the span of h^{d} × l_{d}, l_{d} × h^{d} and h^{d} × h^{d}. If D is divisible by 4, then l_{d}^{2} = l_{0} (cf. Exercise 68.3) and it follows that the idempotents h^{d} × l_{d} and l_{d} × h^{d} + h^{d} × h^{d} are orthogonal and their sum is the fifth (the last) nonzero idempotent in the span of h^{d} × l_{d}, l_{d} × h^{d} and h^{d} × h^{d}. The statement (1) follows now from Lemmas 73.2 and 73.19.

(2): (X, p) ≃ (X, p_{1}) ⊕(X, p_{2}) if and only if (X, p)_{E} ≃ (X, p_{1})_{E} ⊕(X, p_{2})_{E} if and only if p̄ is a disjoint union of p̄_{1} and p̄_{2}.

(3): A correspondence α ∈ Ch_{D+i}(X^{2}) determines an isomorphism (X, p)_{E} → (X, p′)(i)_{E} if and only if p̄ = (h^{0} × h^{i}) · ᾱ and p̄′ = (h^{i} × h^{0}) · ᾱ. □

Corollary 94.2. The motive of any anisotropic smooth projective quadric X decomposes into a direct sum of indecomposable summands. Moreover, such a decomposition is unique and the number of summands coincides with the number of the minimal cycles in \overline{Ch}_{D}(X^{2}), where D = dim X.

Exercise 94.3 (Rost motives). Let π be an anisotropic n-fold Pfister form. Show the following:

(1) The decomposition of the motive of the projective quadric of π into a sum of indecomposable summands has the form ∐_{i=0}^{2^{n−1}−1} R_{π}(i) for some motive R_{π} uniquely determined by π. The motive R_{π} is called the Rost motive associated to π.

(2) For any splitting field extension E/F of π, we have

(R_{π})_{E} ≃ ℤ/2ℤ ⊕ ℤ/2ℤ(2^{n−1} − 1).

(3) The motive of the quadric given by any 1-codimensional subform of π decomposes as ∐_{i=0}^{2^{n−1}−2} R_{π}(i).

(4) Let φ be a (2^{n−1} + 1)-dimensional nondegenerate subform of π. Find a smooth projective quadric X such that the motive of the quadric of φ decomposes as M(X)(1) ⊕ R_{π}. Finally, reprove all this for motives with integral coefficients.

Theorems 93.1 and 94.1 hold in the more general case of motives with integral coefficients. This was done by Vishik in [133]. We note that the integral versions follow from the ℤ/2ℤ-versions by [51] (cf. also [66, Th. 11.2(c)] or [67, Ch. E]).

Appendices

95. Formally real fields

In this section, we review the Artin-Schreier theory of formally real fields. These results and their proofs, may be found in the books by Lam [89] and Scharlau [121].

Let F be a field, P ⊂ F a subset. We say that P is a preordering of F if P satisfies all of the following:

P + P ⊂ P,  P · P ⊂ P,  −1 ∉ P,  and  ∑ F^{2} ⊂ P.

A preordering P of F is called an ordering if, in addition,

F = P ∪ −P.

A field F is called formally real if

D̃(∞⟨1⟩) := {x ∈ F | x is a sum of squares in F }

is a preordering of F, equivalently if −1 is not a sum of squares in F, i.e., the polynomial t_{1}^{2} + ⋯ + t_{n}^{2} has no nontrivial zero over F for any (positive) integer n. Clearly, if F is formally real, then the characteristic of F must be zero. (If char F ≠ 2, then F is not formally real if and only if F = D̃(∞⟨1⟩).) One checks that a preordering is an ordering if and only if it is maximal with respect to set inclusion in the set of preorderings of F. By Zorn’s lemma, maximal preorderings and therefore orderings exist for F if it is formally real. In particular, a field F is formally real if and only if the space of orderings on F,

𝔛(F) := {P | P is an ordering of F }

is not empty. Every P ∈ 𝔛(F) (if any) contains the preordering D̃(∞⟨1⟩). Let P ∈ 𝔛(F) and 0 ≠ x ∈ F. If x ∈ P, then x is called positive (respectively, negative) with respect to P and we write x >_{P} 0 (respectively, x <_{P} 0). Elements that are positive (respectively, negative) with respect to all orderings of F (if any) are called totally positive (respectively, totally negative). In fact, we have

Proposition 95.1 (cf. [89, Th. VIII.1.12] or [121, Cor. 3.1.7]). Suppose that F is formally real. Then D̃(∞⟨1⟩) = ⋂_{P ∈𝔛(F)} P, i.e., a nonzero element of F is totally positive if and only if it is a sum of squares.

It follows that a formally real field has precisely one ordering if and only if D̃(∞⟨1⟩) is an ordering in F, e.g., ℚ or ℝ. The field of real numbers even has ℝ^{2} as an ordering. A formally real field F having F^{2} as an ordering is called euclidean. For such a field every element is either a square or the negative of a square. For example, the field of real constructible numbers is euclidean.

A formally real field is called real closed if it has no proper algebraic extension that is formally real. If F is such a field, then it must be euclidean. Let K/F be an algebraic field extension with K real closed. Then K^{2} ∩ F is an ordering on F.

Let Q ∈ 𝔛(K). The pair (K, Q) is called an ordered field. Let K/F be a field extension with K formally real. If P ∈ 𝔛(F) satisfies P = Q ∩ F, then (K, Q)/(F, P) is called an extension of ordered fields and Q is called an extension of P. If, in addition, K/F is algebraic and there exist no extension (L, R)/(K, Q) with L/K nontrivial algebraic, we call (K, Q) a real closure of (F, P).

Proposition 95.2 (cf. [121, Th. 3.1.14]). If (K, Q) is a real closure of (F, P), then K is real closed and Q = K^{2}.

The key to proving this is Theorem 95.3 (cf. [121, Th. 3.1.9]). Let (F, P) be an ordered field.

(1) Let d ∈ F and K = F (\sqrt{d}). Then there exists an extension of P to K if and only if d ∈ P.

(2) If K/F is finite of odd degree, then there exists an extension of P to K.

The main theorem of Artin-Schreier Theory is

Theorem 95.4 (cf. [89, Th. VIII.2.8] or [121, Th. 3.1.13 and Th. 3.2.8]). Every ordered field (F, P) has a real closure (\overline{F} , \overline{F}^{2}) and this real closure is unique up to a canonical F -isomorphism and this isomorphism is order-preserving.

Because of the last results, if we fix an algebraic closure F̃ of a formally real field F and P ∈ 𝔛(F), then there exists a unique real closure (\overline{F} , \overline{F}^{2}) of (F, P) with \overline{F} ⊂ F̃. We denote \overline{F} by F_{P}.

96. The space of orderings

We view the space of orderings 𝔛(F) on a field F as a subset of the space of functions {±1}^{F^{×}} by the embedding

𝔛(F) → {±1}^{F^{×}} via P ↦ (sign_{P} : x ↦ sign_{P} x)

(the sign of x in F rel P). Giving {±1} the discrete topology, we have {±1}^{F^{×}} is Hausdorff and by Tychonoff’s Theorem, compact. The collection of clopen (i.e., open and closed) sets given by

(96.1)  H_{ε}(a) := {g ∈ {±1}^{F^{×}} | g(a) = −ε}

for a ∈ F^{×} and ε ∈ {±1} forms a subbase for the topology of {±1}^{F^{×}}, hence {±1}^{F^{×}} is also totally disconnected. Consequently, {±1}^{F^{×}} is a boolean space (i.e., a compact totally disconnected Hausdorff space). Let 𝔛(F) have the induced topology arising from the embedding f : 𝔛(F) → {±1}^{F^{×}}.

Theorem 96.2. 𝔛(F) is a boolean space.

Proof. It suffices to show that 𝔛(F) is closed in {±1}^{F^{×}}. Take any element s ∈ {±1}^{F^{×}} \ f(𝔛(F)). First suppose that s is the constant function ε. Then the clopen set H_{ε}(ε) is disjoint from f(𝔛(F)) and contains s, so it separates s from f(𝔛(F)). Assume that s is not a constant function, hence, is surjective. Since s^{−1}(1) is not an ordering on F, there exist a, b ∈ F^{×} such that s(a) = 1 = s(b) (i.e., a, b are “positive”) but either s(a + b) = −1 or s(ab) = −1. Let c = ab if s(ab) = −1, otherwise let c = a + b. As there cannot be an ordering in which a and b are positive but c negative, H_{1}(−a) ∩ H_{1}(−b) ∩ H_{−1}(−c) is disjoint from f(𝔛(F)) and contains s, so it separates s from f(𝔛(F)). □

As we are identifying 𝔛(F) with its image in {±1}^{F^{×}}, we see that the collection of sets

H(a) = H_{F} (a) := H_{1}(a) ⊂ 𝔛(F), a ∈ F^{×},

forms a subbasis of clopen sets for the topology of 𝔛(F) called the Harrison subbasis. So H(a) is the set of orderings on which a is negative. It follows that the collection

97. C_{n}-FIELDS 385

of sets

H(a_{1}, …, a_{n}) = H_{F} (a_{1}, …, a_{n}) := ⋂_{i=1}^{n} H(a_{i}), a_{1}, …, a_{n} ∈ F^{×}

forms a basis for the topology of 𝔛(F).

97. C_{n}-fields

We call a homogeneous polynomial of (total) degree d a d-form. A field F is called a C_{n}-field if every d-form over F in at least d^{n} + 1 variables has a nontrivial zero over F.

For example, a field is algebraically closed if and only if it is a C_{0}-field. Every finite field is a C_{1}-field by the Chevalley-Warning Theorem (cf. [123], I.2, Th. 3).

An n-form in n-variables over F is called a normic form if it has no nontrivial zero. For example, let E/F be a finite field extension of degree n. Let {x_{1}, …, x_{n}} be an F -basis for E. Then the norm form of the extension E/F is the polynomial

N_{E(t_{1},...,t_{n})/F (t_{1},...,t_{n})}(t_{1}x_{1} + ⋯ + t_{n}x_{n})

over F in the variables t_{1}, …, t_{n} is of degree n and has no nontrivial zero, hence it is normic (the reason for the name).

Lemma 97.1. Let F be a nonalgebraically closed field. Then there exist normic forms of arbitrarily large degree.

Proof. There exists a normic form φ of degree n for some n > 1. Having defined a normic form φ_{s} of degree n^{s}, let

φ_{s+1} := φ(φ_{s}|φ_{s}| … |φ_{s}).

This notation means that new variables are to be used after each occurrence of |. The form φ_{s+1} of degree n^{s+1} has no nontrivial zero. □

Theorem 97.2. Let F be a C_{n}-field and let f_{1}, …, f_{r} be d-forms in N common variables. If N > rd^{n}, then the forms have a common nontrivial zero in F.

Proof. Suppose first that n = 0 (i.e., F is algebraically closed) or d = 1. As N > r, it follows from [125, Ch. I, §6.2, Prop.] that the forms have a common nontrivial zero over F.

So we may assume that n > 0 and d > 1. By Lemma 97.1, there exists a normic form φ of degree at least r. We define a sequence of forms φ_{i}, i ≥ 1, of degree d_{i} in N_{i} variables as follows. Let φ_{1} = φ. Assuming that φ_{i} is defined let

φ_{i+1} = φ(f_{1}, …, f_{r} | f_{1}, …, f_{r} | … | f_{1}, …, f_{r} | 0, …, 0),

where zeros occur in < r places. The forms f_{i} between two consecutive signs | have the same sets of variables.

If x ∈ ℝ, let [x] denote the largest integer ≤ x. We have

(97.3)  d_{i+1} = dd_{i} and N_{i+1} = N[\frac{N_{i}}{r} ].

Note that since N > rd^{n} ≥ 2r, we have N_{i} → ∞ as i → ∞.

Set

(97.4)  α_{i} = \frac{r}{N_{i}} [\frac{N_{i}}{r} ].

We have α_{i} → 1 as i → ∞. It follows from (97.3) and (97.4) that

\frac{N_{i+1}}{d_{i+1}^{n}} = \frac{N_{i}}{d_{i}^{n}} · \frac{α_{i}N}{rd^{n}} .

Since N > rd^{n} and α_{i} → 1, there is a β > 1 and an integer s such that \frac{α_{i}N}{rd^{n}} > β if i ≥ s. Therefore, we have

\frac{N_{i+1}}{d_{i+1}^{n}} > \frac{N_{i}}{d_{i}^{n}} · β,

if i ≥ s. It follows that N_{k} > d_{k}^{n} for some k. As F is a C_{n}-field, the form φ_{k} has a nontrivial zero. Choose the smallest k with this property. By definition of φ_{k}, a nontrivial zero of φ_{k} gives rise to a nontrivial common zero of the forms f_{1}, …, f_{r}. □

Corollary 97.5. Let F be a C_{n}-field and let K/F be an algebraic field extension. Then K is a C_{n}-field.

Proof. Let f be a d-form over K in N variables with N > d^{n}. The coefficients of f belong to a finite field extension of F, so we may assume that K/F is a finite extension. Let {x_{1}, …, x_{r}} be an F -basis for K. Choose variables t_{ij}, i = 1, …, N, j = 1, …, r over F and set

t_{i} = t_{i1}x_{1} + ⋯ + t_{ir}x_{r}

for every i. Then

f(t_{1}, …, t_{N}) = f_{1}(t_{ij})x_{1} + ⋯ + f_{r}(t_{ij})x_{r}

for some d-forms f_{j} in rN variables. Since rN > rd^{n}, it follows from Theorem 97.2 that the forms f_{j} have a nontrivial common zero over F which produces a nontrivial zero of f over K. □

Corollary 97.6. Let F be a C_{n}-field. Then F (t) is a C_{n+1}-field.

Proof. Let f be a d-form in N variables over F (t) with N > d^{n+1}. Clearing denominators of the coefficients of f we may assume that all the coefficients are polynomials in t. Choose variables t_{ij}, i = 1, …, N, j = 0, …, m for some m and set

t_{i} = t_{i0} + t_{i1}t + ⋯ + t_{im}t^{m}

for every i. Then

f(t_{1}, …, t_{N}) = f_{0}(t_{ij})t^{0} + ⋯ + f_{dm+r}(t_{ij})t^{dm+r}

for some d-forms f_{j} in N(m+1) variables over F and r = deg_{t}(f). Since N > d^{n+1}, one can choose m such that N(m + 1) > (dm + r + 1)d^{n}. By Theorem 97.2, the forms f_{j} have a nontrivial common zero over F which produces a nontrivial zero of f over F (t). □

Corollaries 97.5 and 97.6 yield

Theorem 97.7. Let F be a C_{n}-field and let K/F be a field extension of transcendence degree m. Then K is a C_{n+m}-field.

As algebraically closed fields are C_{0}-fields, the theorem shows that a field of transcendence degree n over an algebraically closed field is a C_{n}-field. In particular, we have the classical Tsen Theorem:

Theorem 97.8. If F is algebraically closed and K/F is a field extension of transcendence degree 1, then the Brauer group Br(K) is trivial.

Proof. Let A be a central division algebra over K of degree d > 1. The reduced norm form Nrd of D is a form of degree d in d^{2} variables. By Theorem 97.7, K is a C_{1}-field, hence Nrd has a nontrivial zero, a contradiction. □

98. Algebras

For more details see [86] and [52].

98.A. Semisimple, separable and étale algebras. Let F be a field. A finite-dimensional (associative, unital) F -algebra A is called simple if A has no nontrivial (two-sided) ideals. By Wedderburn’s theorem, every simple F -algebra is isomorphic to the matrix algebra 𝕄_{n}(D) for some n and a division F -algebra D uniquely determined by A up to isomorphism over F.

An F -algebra A is called semisimple if A is isomorphic to a (finite) product of simple algebras.

An F -algebra A is called separable if the L-algebra A_{L} := A⊗_{F} L is semisimple for every field extension L/F. This is equivalent to A being a finite product of the matrix algebras 𝕄_{n}(D), where D is a division F -algebra with center a finite separable field extension of F. Separable algebras satisfy the following descent condition: If A is an F -algebra and E/F is a field extension, then A is separable if and only if A_{E} is separable as an E-algebra.

Let A be a finite-dimensional commutative F -algebra. If A is separable, it is called étale. Consequently, A is étale if and only if A is a finite product of finite separable field extensions of F. An étale F -algebra A is called split if A is isomorphic to a product of several copies of F. Let A be a commutative (associative, unital) F -algebra. The determinant (respectively, the trace) of the linear endomorphism of A given by left multiplication by an element a ∈ A is called the norm N_{A}(a) (respectively, the trace Tr_{A}(a)). We have Tr_{A}(a + a′) = Tr_{A}(a) + Tr_{A}(a′) and N_{A}(aa′) = N_{A}(a) N_{A}(a′) for all a, a′ ∈ A. Every a ∈ A satisfies the characteristic polynomial equation

a^{n} − Tr_{A}(a)a^{n−1} + ⋯ + (−1)^{n} N_{A}(a) = 0

where n = dim A.

98.B. Quadratic algebras. A quadratic algebra A over F is an F -algebra of dimension 2. A quadratic algebra is necessarily commutative. Every element a ∈ A satisfies the quadratic equation

(98.1)  a^{2} − Tr_{A}(a)a + N_{A}(a) = 0.

For every a ∈ A, set ā := Tr_{A}(a) − a. We have \overline{aa′} = āā′ for all a, a′ ∈ A. Indeed, since dim A = 2, it suffices to check the equality when a ∈ F or a′ ∈ F (this is obvious) and when a′ = a (it follows from the quadratic equation). Thus the map a ↦ ā is an algebra automorphism of A of exponent 2. We have

Tr_{A}(a) = a + ā and N_{A}(a) = aā.

A quadratic F -algebra A is étale if A is either a quadratic separable field extension of F or A is split, i.e., is isomorphic to F × F.

Let A and B be two quadratic étale F -algebras. The subalgebra A ⋆ B of the tensor product A ⊗_{F} B consisting of all elements stable under the automorphism of A ⊗_{F} B defined by x ⊗ y ↦ x̄ ⊗ ȳ is also a quadratic étale F -algebra. The operation ⋆ on quadratic étale F -algebras yields a (multiplicative) group structure on the set Ét_{2}(F) of isomorphism classes [A] of quadratic étale F -algebras A. Thus [A] · [B] = [A ⋆ B]. Note that Ét_{2}(F) is an abelian group of exponent 2.

Example 98.2. If char F ≠ 2, every quadratic étale F -algebra is isomorphic to

F_{a} := F [t]/(t^{2} − a)

for some a ∈ F^{×}. Let j be the class of t in F_{a}. For every u = x + yj, we have

ū = x − yj, Tr(u) = 2x, and N(u) = x^{2} − ay^{2}.

The assignment a ↦ [F_{a}] gives rise to an isomorphism F^{×}/F^{×2} ≅ Ét_{2}(F).

Example 98.3. If char F = 2, every quadratic étale F -algebra is isomorphic to

F_{a} := F [t]/(t^{2} + t + a)

for some a ∈ F. Let j be the class of t in F_{a}. For every u = x + yj, we have

ū = x + y + yj, Tr(u) = y, and N(u) = x^{2} + xy + ay^{2}.

The assignment a ↦ [F_{a}] induces an isomorphism F/ Im ℘ ≅ Ét_{2}(F), where the Artin-Schreier map ℘ : F → F is defined by ℘(x) = x^{2} + x.

98.C. Brauer group. An F -algebra A is called central if F · 1 coincides with the center of A. A central simple F -algebra A is called split if A ≅ 𝕄_{n}(F) for some n.

Two central simple F -algebras A and B are called Brauer equivalent if 𝕄_{n}(A) ≅ 𝕄_{m}(B) for some n and m. For example, all split F -algebras are Brauer equivalent.

The set Br(F) of all Brauer equivalence classes of central simple F -algebras is a torsion abelian group with respect to the tensor product operation A ⊗_{F} B, called the Brauer group of F. The identity element of Br(F) is the class of split F -algebras.

The class of a central simple F -algebra A will be denoted by [A] and the product of [A] and [B] in the Brauer group, represented by the tensor product A ⊗_{F} B, will be denoted by [A] · [B].

The inverse class of A in Br(F) is given by the class of the opposite algebra A^{op}. The order of [A] in Br(F) is called the exponent of A and will be denoted by exp(A). In particular, exp(A) divides 2 if and only if A^{op} ≅ A, i.e., A has an anti-automorphism.

For an integer m, we write Br_{m}(F) for the subgroup of all classes [A] ∈ Br(F) such that [A]^{m} = 1.

Let A be a central simple algebra over F and let L/F be a field extension. Then A_{L} is a central simple algebra over L. (In particular, every central simple F -algebra is separable.) The correspondence [A] ↦ [A_{L}] gives rise to a group homomorphism

r_{L/F} : Br(F) → Br(L).

We set Br(L/F) := Ker r_{L/F}. The class A is said to be split over L (and L/F is called a splitting field extension of A) if the algebra A_{L} is split, equivalently [A] ∈ Br(L/F).

A central simple F -algebra A is isomorphic to 𝕄_{k}(D) for a central division F - algebra D, unique up to isomorphism. The integers \sqrt{dim D} and \sqrt{dim A} are called the index and the degree of A respectively and denoted by ind(A) and deg(A).

Fact 98.4. Let A be a central simple algebra over F and L/F a finite field extension. Then

ind(A_{L}) | ind(A) | ind(A_{L}) · [L : F].

Corollary 98.5. Let A be a central simple algebra over F and L/F a finite field extension. Then

(1) If L is a splitting field of A, then ind(A) divides [L : F].

(2) If [L : F] is relatively prime to ind(A), then ind(A_{L}) = ind(A).

Fact 98.6. Let A be a central division algebra over F.

(1) A subfield K ⊂ A is maximal if and only if [K : F] = ind(A). In this case K is a splitting field of A.

(2) Every splitting field of A of degree ind(A) over F can be embedded into A over F as a maximal subfield.

98.D. Severi-Brauer varieties. Let A be a central simple F -algebra of degree n. Let r be an integer dividing n. The generalized Severi-Brauer variety SB_{r}(A) of A is the variety of right ideals of dimension rn in A [86, 1.16]. We simply write SB(A) for SB_{1}(A).

If A is split, i.e., A = End(V) for a vector space V of dimension n, every right ideal I in A of dimension rn has the form I = Hom(V, U) for a uniquely determined subspace U ⊂ V of dimension r. Thus the correspondence I ↦ U yields an isomorphism SB_{r}(A) ≅ Gr_{r}(V), where Gr_{r}(V) is the Grassmannian variety of r-dimensional subspaces in V. In particular, SB(A) ≅ ℙ(V).

Proposition 98.7 ([86, Prop. 1.17]). Let A be a central simple F -algebra and let r be an integer dividing deg(A). Then the Severi-Brauer variety X = SB_{r}(A) has a rational point over an extension L/F if and only if ind(A_{L}) divides r. In particular, SB(A) has a rational point over L if and only if A is split over L.

Let V_{1} and V_{2} be vector spaces over F of finite dimension. The Segre closed embedding is the morphism

ℙ(V_{1}) × ℙ(V_{2}) → ℙ(V_{1} ⊗_{F} V_{2})

taking a pair of lines U_{1} and U_{2} in V_{1} and V_{2} respectively to the line U_{1} ⊗_{F} U_{2} in V_{1} ⊗_{F} V_{2}.

Example 98.8. The Segre embedding identifies ℙ_{F}^{1} ×ℙ_{F}^{1} with a projective quadric in ℙ_{F}^{3} .

The Segre embedding can be generalized as follows. Let A_{1} and A_{2} be two central simple algebras over F. Then the correspondence (I_{1}, I_{2}) ↦ I_{1} ⊗ I_{2} yields a closed embedding

SB(A_{1}) × SB(A_{2}) → SB(A_{1} ⊗_{F} A_{2}).

98.E. Quaternion algebras. Let L/F be a Galois quadratic field extension with Galois group {e, g} and b ∈ F^{×}. The F -algebra Q = (L/F, b) := L ⊕ Lj, where the symbol j satisfies j^{2} = b and jl = g(l)j for all l ∈ L, is central simple of dimension 4 and is called a quaternion algebra. We have Q is either split, i.e., isomorphic to the matrix algebra 𝕄_{2}(F), or a division algebra. The algebra Q carries a canonical involution : Q → Q satisfying j̄ = −j and l̄ = g(l) for all l ∈ L.

Using the canonical involution, we define the linear reduced trace map as

Trd : Q → F by Trd(q) = q + q̄,

and the quadratic reduced norm map as

Nrd : Q → F by Nrd(q) = q · q̄.

An element q ∈ Q is called a pure quaternion if Trd(q) = 0, or equivalently, q̄ = −q. Denote by Q′ the 3-dimensional subspace of all pure quaternions. We have Nrd(q) = −q^{2} for any q ∈ Q′.

Proposition 98.9. Every central division algebra of dimension 4 is isomorphic to a quaternion algebra.

Proof. Let L ⊂ Q be a separable quadratic subfield. By the Skolem-Noether Theorem, the only nontrivial automorphism g of L over F extends to an inner automorphism of Q, i.e., there is j ∈ Q^{×} such that jlj^{−1} = g(l) for all l ∈ L. Clearly, Q = L⊕Lj and j^{2} commutes with j and L. Hence j^{2} belongs to the center of Q, i.e., b := j^{2} ∈ F^{×}. Therefore, Q is isomorphic to the quaternion algebra Q = (L/F, b). □

Example 98.10. If char F ≠ 2, a separable quadratic subfield L of a quaternion algebra Q = (L/F, b) is of the form L = F (i) with i^{2} = a ∈ F^{×}. Hence Q has a basis {1, i, j, k := ij} with multiplication table

i^{2} = a, j^{2} = b, ji + ij = 0,

for some b ∈ F^{×}. We shall denote the algebra generated by i and j with these relations by \binom{a,b}{F}.

The space of pure quaternions has {i, j, k} as a basis. For every q = x + yi + zj + wk with x, y, z, w ∈ F, we have

q̄ = x − yi − zj − wk, Trd(q) = 2x, and Nrd(q) = x^{2} − ay^{2} − bz^{2} + abw^{2}.

Example 98.11. If char F = 2, a separable quadratic subfield L of a quaternion algebra Q = (L/F, b) is of the form L = F (s) with s^{2} + s + c = 0 for some c ∈ F. Set i = sj/b. We have i^{2} = a := c/b. Hence Q has a basis {1, i, j, k := ij} with the multiplication table

(98.12)  i^{2} = a, j^{2} = b, ji + ij = 1.

We shall denote by \left[{a,b \atop F}\right] the algebra given by the generators i and j and the relations (98.12). Note that this is also a quaternion algebra (in fact split) when b = 0.

The space of pure quaternions has {1, i, j} as a basis. For every q = x + yi + zj + wk with x, y, z, w ∈ F, we have

q̄ = (x + w) + yi + zj + wk, Trd(q) = w,
and Nrd(q) = x^{2} + ay^{2} + bz^{2} + abw^{2} + xw + yz.

The classes of quaternion F -algebras satisfy the following relations in Br(F):

Fact 98.13. Suppose that char F ≠ 2. Then

(1) \binom{aa′,b}{F} = \binom{a,b}{F} · \binom{a′,b}{F} and \binom{a,bb′}{F} = \binom{a,b}{F} · \binom{a,b′}{F}.

(2) \binom{a,b}{F} = \binom{b,a}{F}.

(3) \binom{a,b}{F}^{2} = 1.

(4) \binom{a,b}{F} = 1 if and only if a is a norm of the quadratic étale extension F_{b}/F.

Fact 98.14. Suppose that char F = 2. Then

(1) \left[{a+a′,b \atop F}\right] = \left[{a,b \atop F}\right] · \left[{a′,b \atop F}\right] and \left[{a,b+b′ \atop F}\right] = \left[{a,b \atop F}\right] · \left[{a,b′ \atop F}\right].

(2) \left[{ab,c \atop F}\right] · \left[{bc,a \atop F}\right] · \left[{ca,b \atop F}\right] = 1.

(3) \left[{a,b \atop F}\right] = \left[{b,a \atop F}\right].

(4) \left[{a,b \atop F}\right]^{2} = 1.

(5) \left[{a,b \atop F}\right] = 1 if and only if a is a norm of the quadratic étale extension F_{ab}/F.

We shall need the following properties of quaternion algebras.

Lemma 98.15 (Chain Lemma). Let \binom{a,b}{F} and \binom{c,d}{F} be isomorphic quaternion algebras over a field F of characteristic not 2. Then there is an e ∈ F^{×} satisfying \binom{a,b}{F} ≃ \binom{a,e}{F} ≃ \binom{c,e}{F} ≃ \binom{c,d}{F}.

Proof. Note that if x and y are pure quaternions in a quaternion algebra Q that are orthogonal with respect to the reduced trace bilinear form, i.e., Trd(xy) = 0, then Q ≃ \binom{x^{2},y^{2}}{F}. Let Q = \binom{a,b}{F}. By assumption, there are pure quaternions

x, y satisfying x^{2} = a and y^{2} = c. Choose a pure quaternion z orthogonal to x and y. Setting e = z^{2}, we have Q ≃ \binom{a,e}{F} ≃ \binom{c,e}{F}. □

Lemma 98.16. Let Q be a quaternion algebra over a field F of characteristic 2. Suppose that Q is split by a purely inseparable field extension K/F such that K^{2} ⊂ F. Then Q ≅ \left[{a,b \atop F}\right] with a ∈ K^{2}.

Proof. First suppose that K = F (\sqrt{a}) is a quadratic extension of F. By Fact 98.6, we know that K can be embedded into Q. Therefore there exists an i ∈ Q \ F such that i^{2} = a ∈ K^{2}. Note that i is a pure quaternion in Q′ \ F. The linear map Q′ → F taking an x to ix + xi is nonzero, hence there is a j ∈ Q′ such that ij + ji = 1. Hence, Q ≅ \left[{a,b \atop F}\right] where b = j^{2}.

In the general case, write Q = \left[{c,d \atop F}\right]. By Property (5) of Fact 98.14, we have

c = x^{2} +xy +cdy^{2} for some x, y ∈ K. Since x^{2} and y^{2} belong to F, we have xy ∈ F. Hence the extension E = F (x, y) splits Q and [E : F] ≤ 2. The statement now follows from the first part of the proof. □

Let σ be an automorphism of a ring R. Denote by R[t, σ] the ring of σ-twisted polynomials in the variable t with multiplication defined by tr = σ(r)t for all r ∈ R. For example, if σ is the identity, then R[t, σ] is the ordinary polynomial ring R[t] over R. Observe that if R has no zero divisors, then neither does R[t, σ].

Example 98.17. Let A be a central division algebra over a field F. Consider an automorphism σ of the polynomial ring A[x] defined by σ(a) = a for all a ∈ A and

σ(x) = \{ \begin{matrix}−x & if char F ≠ 2, \\ x + 1 & if char F = 2. \end{matrix}

Let B be the quotient ring of A[x][t, σ]. The ring B is a division algebra over its center E where

E = \{ \begin{matrix}F (x^{2}, t^{2}) & if char F ≠ 2, \\ F (x^{2} + x, t^{2}) & if char F = 2. \end{matrix}

Moreover, B = A ⊗_{F} Q, where Q is a quaternion algebra over E satisfying

Q = \{ \begin{matrix}\binom{x^{2},t^{2}}{E} & if char F ≠ 2, \\ \left[{(x^{2}+x)/t^{2},t^{2} \atop E}\right] & if char F = 2. \end{matrix}

Iterating the construction in Example 98.17 yields the following

Proposition 98.18. For any field F and integer n ≥ 1, there is a field extension L/F and a central division L-algebra that is a tensor product of n quaternion algebras.

We now study interactions between two quaternion algebras.

Theorem 98.19. Let Q_{1} and Q_{2} be division quaternion algebras over F. Then the following conditions are equivalent:

(1) The tensor product Q_{1} ⊗_{F} Q_{2} is not a division algebra.

(2) Q_{1} and Q_{2} have isomorphic separable quadratic subfields.

(3) Q_{1} and Q_{2} have isomorphic quadratic subfields.

Proof. (1) ⇒ (2): Write X_{1}, X_{2}, and X for Severi-Brauer varieties of Q_{1}, Q_{2}, and A := Q_{1} ⊗_{F} Q_{2} respectively. The morphism X_{1} × X_{2} → X taking a pair of ideals I_{1} and I_{2} to the ideal I_{1} ⊗ I_{2} identifies X_{1} × X_{2} with a twisted form of a 2-dimensional quadric in X (cf. §98.D).

Let Y be the generalized Severi-Brauer variety of rank 8 ideals in A. A rational point of Y, i.e., a right ideal J ⊂ A of dimension 8, defines the closed curve C_{J} in X comprising all ideals of rank 4 contained in J. In the split case, Y is the Grassmannian variety of planes and C_{J} is the projective line (the projective space of the plane corresponding to J) properly intersecting the quadric X_{1} × X_{2} in two points. Thus there is a nonempty open subset U ⊂ Y with the following property: for any rational point J ∈ U, we have C_{J} ∩ (X_{1} × X_{2}) = {x}, where x is a point of degree 2 with residue field L a separable quadratic field extension of F. By assumption, there is a right ideal I ⊂ A of dimension 8, i.e., Y (F) ≠ ∅. The algebraic group G of invertible elements of A acts transitively on Y, i.e., the morphism G → Y taking an a to the ideal aI is surjective. As rational points of G are dense in G, we have rational points of Y are dense in Y. Hence U possesses a rational point J.

As X_{1}(L) × X_{2}(L) = (X_{1} × X_{2})(L) ≠ ∅, it follows that the field L splits both Q_{1} and Q_{2} and therefore L is isomorphic to quadratic subfields in Q_{1} and Q_{2}.

(2) ⇒ (3) is trivial.

(3) ⇒ (1): Let L/F be a common quadratic subfield of both Q_{1} and Q_{2}. It follows that Q_{1} and Q_{2} and, hence, A are split by L. It follows from Corollary 98.5 that ind(A) ≤ 2, i.e., A is not a division algebra. □

99. Galois cohomology

For more details see [124].

99.A. Galois modules and Galois cohomology groups. Let Γ be a profinite group and let M be a (left) discrete Γ-module. For any n ∈ ℤ, let H^{n}(Γ, M) denote the nth cohomology group of Γ with coefficients in M. In particular, the group H^{n}(Γ, M) is trivial, if n < 0 and

H^{0}(Γ, M) = M^{Γ} := {m ∈ M | γm = m for all γ ∈ Γ},

the subgroup of Γ-invariant elements of M.

An exact sequence 0 → M′ → M → M′′ → 0 gives rise to an infinite long exact sequence of cohomology groups

0 → H^{0}(Γ, M′) → H^{0}(Γ, M) → H^{0}(Γ, M′′) → H^{1}(Γ, M′) → H^{1}(Γ, M) → ….

Let F be a field. Denote by Γ_{F} the absolute Galois group of F, i.e., the Galois group of a separable closure F_{sep} of the field F over F. A discrete Γ_{F} -module is called a Galois module over F. For a Galois module M over F, we write H^{n}(F, M) for H^{n}(Γ_{F} , M) .

Example 99.1. (1) Every abelian group A can be viewed as a Galois module over F with trivial action. We have H^{0}(F, A) = A and H^{1}(F, A) = Hom_{c}(Γ_{F} , A), the group of continuous homomorphisms (where A is viewed with discrete topology). In particular, H^{1}(F, A) is trivial if A is torsion-free, e.g., H^{1}(F, ℤ) = 0.

The group H^{1}(F, ℚ/ℤ) = Hom_{c}(Γ_{F} , ℚ/ℤ) is called the character group of Γ_{F} and will be denoted by χ(Γ_{F}).

The cohomology group H^{n}(F, M) is torsion for every Galois module M and any n ≥ 1.

Since the group ℚ is uniquely divisible, we have H^{n}(F, ℚ) = 0 for all n ≥ 1. The cohomology exact sequence for the short exact sequence of Galois modules with trivial action

0 → ℤ → ℚ → ℚ/ℤ → 0

then gives an isomorphism H^{n}(F, ℚ/ℤ) →^{∼} H^{n+1}(F, ℤ) for any n ≥ 1. In particular, H^{2}(F, ℤ) ≅ χ(Γ_{F}).

Let m be a natural integer. The cohomology exact sequence for the short exact sequence

0 → ℤ ⟶^{m} ℤ → ℤ/mℤ → 0

gives an isomorphism of H^{1}(F, ℤ/mℤ) with the subgroup χ_{m}(Γ_{F}) of characters of exponent m.

(2) The cohomology groups H^{n}(F, F_{sep}) with coefficients in the additive group F_{sep} are trivial if n > 0. If char F = p > 0, the cohomology exact sequence for the short exact sequence

0 → ℤ/pℤ → F_{sep} ⟶^{℘} F_{sep} → 0,

where ℘ is the Artin-Schreier map defined by ℘(x) = x^{p} − x, yields canonical isomorphisms

H^{n}(F, ℤ/pℤ) ≅ \{ \begin{matrix}ℤ/pℤ & if n = 0, \\ F/℘(F) & if n = 1, \\ 0 & otherwise. \end{matrix}

In fact, H^{n}(F, M) = 0 for all n ≥ 2 and every Galois module M over F of characteristic p satisfying pM = 0.

(3) We have the following canonical isomorphisms for the cohomology groups with coefficients in the multiplicative group F_{sep}^{×}:

H^{n}(F, F_{sep}^{×}) ≅ \{ \begin{matrix}F^{×} & if n = 0, \\ 1 & if n = 1 (Hilbert Theorem 90), \\ Br(F) & if n = 2. \end{matrix}

(4) The group μ_{m} = μ_{m}(F_{sep}) of mth roots of unity in F_{sep} is a Galois submodule of F_{sep}^{×}. We have the following exact sequence of Galois modules:

(99.2)  1 → μ_{m} → F_{sep}^{×} → F_{sep}^{×} → F_{sep}^{×}/F_{sep}^{×m} → 1,

where the middle homomorphism takes x to x^{m}.

If m is not divisible by char F, we have F_{sep}^{×}/F_{sep}^{×m} = 1. Therefore, the cohomology exact sequence (99.2) yields isomorphisms

H^{n}(F, μ_{m}) ≅ \{ \begin{matrix}μ_{m}(F), & if n = 0, \\ F^{×}/F^{×m}, & if n = 1, \\ Br_{m}(F), & if n = 2. \end{matrix}

For any a ∈ F^{×}, we shall write (a)_{m} or simply (a) for the element of H^{1}(F, μ_{m}) corresponding to the coset aF^{×m} in F^{×}/F^{×m}.

If p = char F > 0, we have μ_{p}(F_{sep}) = 1 and the cohomology exact sequence (99.2) gives an isomorphism

H^{1}(F, F_{sep}^{×}/F_{sep}^{×p}) ≅ Br_{p}(F).

Example 99.3. Let ξ ∈ χ_{2}(Γ_{F}) be a nontrivial character. Then Ker(ξ) is a subgroup of Γ_{F} of index 2. By Galois theory, it corresponds to a Galois quadratic field extension F_{ξ}/F. The correspondence ξ ↦ F_{ξ} gives rise to an isomorphism

χ_{2}(Γ_{F}) →^{∼} Ét_{2}(F).

99.B. Cup-products. Let M and N be Galois modules over F. Then the tensor product M ⊗_{ℤ} N is also a Galois module via the diagonal Γ_{F} -action. There is a pairing

H^{m}(F, M) ⊗ H^{n}(F, N) → H^{m+n}(F, M ⊗_{ℤ} N), α ⊗ β ↦ α ∪ β

called the cup-product. When m = n = 0 the cup-product coincides with the natural homomorphism M^{Γ_{F}} ⊗ N^{Γ_{F}} → (M ⊗_{ℤ} N)^{Γ_{F}}.

Fact 99.4. (Cf. [24, Ch. IV, §7].) Let 0 → M′ → M → M′′ → 0 be an exact sequence of Galois modules over F. Suppose that for a Galois module N the sequence

0 → M′ ⊗_{ℤ} N → M ⊗_{ℤ} N → M′′ ⊗_{ℤ} N → 0

is exact. Then the diagram

H^{n}(F, M′′) ⊗ H^{m}(F, N) ⟶^{∪} H^{n+m}(F, M′′ ⊗_{ℤ} N)
∂⊗id↓  ↓^{∂}
H^{n+1}(F, M′) ⊗ H^{m}(F, N) ⟶^{∪} H^{n+m+1}(F, M′ ⊗_{ℤ} N)

is commutative.

Example 99.5. The cup-product

H^{0}(F, F_{sep}^{×}) ⊗ H^{2}(F, ℤ) → H^{2}(F, F_{sep}^{×})

yields a pairing

F^{×} ⊗ Ét(F) → Br_{2}(F).

If char F ≠ 2, we have a∪[F_{b}] = \binom{a,b}{F} for all a, b ∈ F^{×}. In the case that char F = 2,

we have a ∪ [F_{ab}] = \left[{a,b \atop F}\right] for all a ∈ F^{×} and b ∈ F.

Suppose that char F ≠ 2. Then μ_{2} ≃ ℤ/2ℤ. The cup-product

H^{1}(F, ℤ/2ℤ) ⊗ H^{1}(F, ℤ/2ℤ) → H^{2}(F, ℤ/2ℤ)

gives rise to a pairing

F^{×}/F^{×}^{2} ⊗ F^{×}/F^{×}^{2} → Br_{2}(F).

We have (a) ∪ (b) = \binom{a,b}{F} for all a, b ∈ F^{×}. In particular, (a) ∪ (1 − a) = 0 for every a ≠ 0, 1 by Fact 98.13(4).

99.C. Restriction and corestriction homomorphisms. Let M be a Galois module over F and K/F an arbitrary field extension. Separable closures of F and K can be chosen so that F_{sep} ⊂ K_{sep}. The restriction then yields a continuous group homomorphism Γ_{K} → Γ_{F}. In particular, M has the structure of a discrete Γ_{K}-module and we have the restriction homomorphism

r_{K/F} : H^{n}(F, M) → H^{n}(K, M).

If K/F is a finite separable field extension, then Γ_{K} is an open subgroup of finite index in Γ_{F}. For every n ≥ 0 there is a natural corestriction homomorphism

c_{K/F} : H^{n}(K, M) → H^{n}(F, M).

In the case n = 0, the map c_{K/F} : M^{Γ_{K}} → M^{Γ_{F}} is given by x → ∑ γ(x) where the sum is over a left transversal of Γ_{K} in Γ_{F}. The composition c_{K/F} ∘ r_{K/F} is multiplication by [K : F].

Let K/F be an arbitrary finite field extension and let M be a Galois module over F. Let E/F be the maximal separable sub-extension in K/F. As the restriction map Γ_{K} → Γ_{E} is an isomorphism, we have a canonical isomorphism s : H^{n}(K, M) →^{∼} H^{n}(E, M). We define the corestriction homomorphism c_{K/F} : H^{n}(K, M) → H^{n}(F, M) as [K : E] times the composition c_{E/F} ∘ s.

Example 99.6. The corestriction homomorphism c_{K/F} : H^{1}(K, μ_{m}) → H^{1}(F, μ_{m}) takes a class (x)_{m} to (N_{K/F} (x))_{m}.

Example 99.7. The restriction map in Galois cohomology agrees with the restriction map for Brauer groups defined in §98.C. The corestriction in Galois cohomology yields a map c_{K/F} : Br(K) → Br(F) for a finite field extension K/F. Since the composition c_{K/F} ∘ r_{K/F} is the multiplication by m = [K : F] we have Br(K/F) ⊂ Br_{m}(F).

Let K/F be a finite separable field extension and let M be a Galois module over K. We view Γ_{K} as a subgroup of Γ_{F}. Denote by Ind_{K/F} (M) the group Map_{Γ_{K}} (Γ_{F} , M) of Γ_{K}-equivariant maps Γ_{F} → M, i.e., maps f : Γ_{F} → M satisfying f(ρδ) = ρf(δ) for all ρ ∈ Γ_{K} and δ ∈ Γ_{F}. The group Ind_{K/F} (M) has a structure of Galois module over F defined by (γf)(δ) = f(δγ) for all f ∈ Ind_{K/F} (M) and γ, δ ∈ Γ_{F}. Consider the Γ_{K}-module homomorphisms

M ⟶^{u} Ind_{K/F} (M) ⟶^{v} M

defined by v(f) = f(1) and

u(m)(γ) = \{ \begin{matrix}m & if γ ∈ Γ_{K}, \\ 0 & otherwise. \end{matrix}

Fact 99.8. Let M be a Galois module over F and let K/F be a finite separable field extension. Then the compositions

H^{n}(F, Ind_{K/F} (M)) ⟶^{r_{K/F}} H^{n}(K, Ind_{K/F} (M)) ⟶^{H^{n}(K,v)} H^{n}(K, M),

H^{n}(K, M) ⟶^{H^{n}(K,u)} H^{n}(K, Ind_{K/F} (M)) ⟶^{c_{K/F}} H^{n}(F, Ind_{K/F} (M))

are isomorphisms inverse to each other.

Suppose, in addition, that M is a Galois module over F. Consider the Γ_{F} - module homomorphisms

(99.9)  0 → M ⟶^{w} Ind_{K/F} (M) ⟶^{t} M → 0

defined by w(m)(γ) = γm and

t(f) = ∑ γ(f(γ^{−1})),

where the sum is taken over a left transversal of Γ_{K} in Γ_{F}.

Corollary 99.10. (1) The composition

H^{n}(F, M) ⟶^{H^{n}(F,w)} H^{n}(F, Ind_{K/F} (M)) →^{∼} H^{n}(K, M)

coincides with r_{K/F}.

(2) The composition

H^{n}(K, M) →^{∼} H^{n}(F, Ind_{K/F} (M)) ⟶^{H^{n}(F,t)} H^{n}(K, M)

coincides with c_{K/F}.

99.D. Residue homomorphism. Let m be an integer. A Galois module M over F is said to be m-periodic if mM = 0. If m is not divisible by char F, we write M(−1) for the Galois module Hom(μ_{m}, M) with the action of Γ_{F} given by (γf)(ξ) = γf(γ^{−1}ξ) for every f ∈ M(−1) (the construction is independent of the choice of m). For example, μ_{m}(−1) = ℤ/mℤ.

Let L be a field with a discrete valuation v and residue field F. Suppose that the inertia group of an extension of v to L_{sep} acts trivially on M. Then M has a natural structure of a Galois module over F.

Fact 99.11 (Cf. [46, §7]). Let L be a field with a discrete valuation v and residue field F. Let M be an m-periodic Galois module L with m not divisible by char F such that the inertia group of an extension of v to L_{sep} acts trivially on M. Then there exists a residue homomorphism

∂_{v} : H^{n+1}(L, M) → H^{n}(F, M(−1))

satisfying (1) If M = μ_{m} and n = 0, then ∂_{v}((x)_{m}) = v(x) + mℤ for every x ∈ L^{×}.

100. MILNOR K-THEORY OF FIELDS 397

(2) For every x ∈ L^{×} with v(x) = 0, we have ∂_{v}(α ∪ (x)_{m}) = ∂_{v}(α) ∪ (x̄)_{m},

where α ∈ H^{n+1}(L, M) and x̄ ∈ F^{×} is the residue of x.

Let X be a variety (integral scheme) over F and let x ∈ X be a regular point of codimension 1. The local ring O_{X,x} is a discrete valuation ring with quotient field F (X) and residue field F (x). For any m-periodic Galois module M over F let

∂_{x} : H^{n+1}(F (X), M) → H^{n}(F (x), M(−1))

denote the residue homomorphism ∂_{v} of the associated discrete valuation v on F (X).

Fact 99.12 (Cf. [46, Th. 9.2]). For every field F, the sequence

0 → H^{n+1}(F, M) ⟶^{r} H^{n+1}(F (t), M)

⟶^{(∂_{x})} ∐_{x∈ℙ^{1}} H^{n}(F (x), M(−1)) ⟶^{c} H^{n}(F, M(−1)) → 0,

where c is the direct sum of the corestriction homomorphisms c_{F (x)/F}, is exact.

99.E. A long exact sequence. Let K = F (\sqrt{a}) be a quadratic field extension of a field F of characteristic not 2. Let M be a 2-periodic Galois module over F.

We have the exact sequence (99.9) of Galois modules over F. By Corollary 99.10, the induced exact sequence of Galois cohomology groups reads as follows:

… ⟶^{∂} H^{n}(F, M) ⟶^{r_{K/F}} H^{n}(K, M) ⟶^{c_{K/F}} H^{n}(F, M) ⟶^{∂} H^{n+1}(F, M) → ….

We now compute the connecting homomorphisms ∂. If n = 0 and M = ℤ/2ℤ, we have the exact sequence

ℤ/2ℤ ⟶^{0} ℤ/2ℤ ⟶^{∂} F^{×}/F^{×2} → K^{×}/K^{×2}.

The kernel of the last homomorphism is the cyclic group {1, (a)}. It follows that ∂(1 + 2ℤ) = (a). By Fact 99.4, the homomorphisms ∂ : H^{n}(F, M) → H^{n+1}(F, M) coincide with the cup-product by (a).

We have proven

Theorem 99.13. Let K = F (\sqrt{a}) be a quadratic field extension of a field F of characteristic not 2 and let M be a 2-periodic Galois module over F. Then the following sequence

… ⟶^{∪(a)} H^{n}(F, M) ⟶^{r_{K/F}} H^{n}(K, M) ⟶^{c_{K/F}} H^{n}(F, M) ⟶^{∪(a)} H^{n+1}(F, M) ⟶^{r_{K/F}} …

is exact.

100. Milnor K-theory of fields

A more detailed exposition on the Milnor K-theory of fields is available in [43, Ch. IX] and [47, Ch. 7].

100.A. Definition. Let F be a field. Let T_{∗} denote the tensor ring of the multiplicative group F^{×}. That is a graded ring with T_{n} the nth tensor power of F^{×} over ℤ. For instance, T_{0} = ℤ, T_{1} = F^{×}, T_{2} = F^{×} ⊗_{ℤ} F^{×}, etc. The graded Milnor ring K_{∗}(F) of F is the factor ring of T_{∗} by the ideal generated by tensors of the form a ⊗ b with a + b = 1.

The class of a tensor a_{1} ⊗a_{2} ⊗…⊗a_{n} in K_{∗}(F) is denoted by {a_{1}, a_{2}, …, a_{n}}_{F} or simply by {a_{1}, a_{2}, …, a_{n}} and is called a symbol. We have K_{n}(F) = 0 if n < 0, K_{0}(F) = ℤ, K_{1}(F) = F^{×}. For n ≥ 2, K_{n}(F) is generated (as an abelian group) by the symbols {a_{1}, a_{2}, …, a_{n}} with a_{i} ∈ F^{×} that are subject to the following defining relations:

(M1) (Multilinearity)

{a_{1}, …, a_{i}a′_{i}, …, a_{n}} = {a_{1}, …, a_{i}, …, a_{n}} + {a_{1}, …, a′_{i}, …, a_{n}};

(M2) (Steinberg Relation) {a_{1}, a_{2}, …, a_{n}} = 0 if a_{i} + a_{i+1} = 1 for some i ∈ [1, n − 1].

Note that the operation in the group K_{n}(F) is written additively. In particular, {ab} = {a} + {b} in K_{1}(F) where a, b ∈ F^{×}.

The product in the ring K_{∗}(F) is given by the rule

{a_{1}, a_{2}, …, a_{n}} · {b_{1}, b_{2}, …, b_{m}} = {a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}, …, b_{m}}.

Fact 100.1. Let a_{1}, …, a_{n} ∈ F^{×}.

(1) For a permutation σ ∈ S_{n}, we have

{a_{σ(1)}, a_{σ(2)}, …, a_{σ(n)}} = sgn(σ){a_{1}, a_{2}, …, a_{n}}.

(2) {a_{1}, a_{2}, …, a_{n}} = 0 if a_{i} + a_{j} = 0 or 1 for some i ≠ j.

A field homomorphism F → L induces the restriction graded ring homomorphism r_{L/F} : K_{∗}(F) → K_{∗}(L) taking a symbol {a_{1}, a_{2}, …, a_{n}}_{F} to the symbol {a_{1}, a_{2}, …, a_{n}}_{L}. In particular, K_{∗}(L) has a natural structure of a left and right graded K_{∗}(F)-module. The image r_{L/F} (α) of an element α ∈ K_{∗}F is also denoted by α_{L}.

If E/L is another field extension, then r_{E/F} = r_{E/L} ∘ r_{L/F}. Thus, K_{∗} is a functor from the category of fields to the category of graded rings.

Proposition 100.2. Let L/F be a quadratic field extension. Then

K_{n}(L) = r_{L/F} (K_{n−1}(F)) · K_{1}(L)

for every n ≥ 1, i.e., K_{∗}(L) is generated by K_{1}(L) as a left K_{∗}(F)-module.

Proof. It is sufficient to treat the case n = 2. Let x, y ∈ L \ F. If x = cy for some c ∈ F^{×}, then {x, y} = {−c, y} ∈ r_{L/F} (K_{1}(F)) · K_{1}(L). Otherwise, as x, y, and 1 are linearly dependent over F, there are a, b ∈ F^{×} such that ax + by = 1. We have

0 = {ax, by} = {x, y} + {x, b} + {a, by},

hence {x, y} = {b}_{L} · {x} − {a}_{L} · {by} ∈ r_{L/F} (K_{1}(F)) · K_{1}(L). □

We write k_{∗}(F) for the graded ring K_{∗}(F)/2K_{∗}(F). Abusing notation, if {a_{1}, …, a_{n}} is a symbol in K_{n}(F), we shall also write it for its coset {a_{1}, …, a_{n}} + 2K_{n}(F) in k_{n}(F).

We need some relations among symbols in k_{2}(F).

Lemma 100.3. We have the following relations in k_{2}(F):

(1) {a, x^{2} − ay^{2}} = 0 for all a ∈ F^{×}, x, y ∈ F satisfying x^{2} − ay^{2} ≠ 0.
(2) {a, b} = {a + b, ab(a + b)} for all a, b ∈ F^{×} satisfying a + b ≠ 0.

Proof. (1): The statement follows from Fact 100.1 if x = 0. Suppose x ≠ 0. By the Steinberg relation, we have

0 = {a(yx^{−1})^{2}, 1 − a(yx^{−1})^{2}} = {a, x^{2} − ay^{2}}.

(2): Since a(a + b) + b(a + b) is a square, by (1) we have

0 = {a(a + b), b(a + b)} = {a, b} + {a + b, ab(a + b)}.  □

100.B. Residue homomorphism. Let L be a field with a discrete valuation v and residue field F. The homomorphism L^{×} → ℤ given by the valuation can be viewed as a homomorphism K_{1}(L) → K_{0}(F). More generally, for every n ≥ 0, there is the residue homomorphism

∂_{v} : K_{n+1}(L) → K_{n}(F)

uniquely determined by the following condition:

If a_{0}, a_{1}, …, a_{n} ∈ L^{×} satisfying v(a_{i}) = 0 for all i = 1, 2, …, n, then

∂_{v}({a_{0}, a_{1}, …, a_{n}}) = v(a_{0}) · {ā_{1}, …, ā_{n}},

where ā ∈ F denotes the residue of a.

Fact 100.4. Let L be a field with a discrete valuation v and residue field F.

(1) If α ∈ K_{∗}(L) and a ∈ L^{×} satisfies v(a) = 0, then

∂_{v}(α · {a}) = ∂_{v}(α) · {ā} and ∂_{v}({a} · α) = −{ā} · ∂_{v}(α).

(2) Let K/L be a field extension and let u be a discrete valuation of K extending v with residue field E. Let e denote the ramification index. Then for every α ∈ K_{∗}(L),

∂_{u}(r_{K/L}(α)) = e · r_{E/F} (∂_{v}(α)).

100.C. Milnor’s theorem. Let X be a variety (integral scheme) over F and let x ∈ X be a regular point of codimension 1. The local ring O_{X,x} is a discrete valuation ring with quotient field F (X) and residue field F (x). Denote by

∂_{x} : K_{∗+1}(F (X)) → K_{∗}(F (x))

the residue homomorphism of the associated discrete valuation on F (X).

The following description of the K-groups of the function field F (t) = F (𝔸_{F}^{1} ) of the affine line is known as Milnor’s Theorem.

Fact 100.5. (Milnor’s Theorem) For every field F, the sequence

0 → K_{n+1}(F) ⟶^{r_{F (t)/F}} K_{n+1}(F (t)) ⟶^{(∂_{x})} ∐_{x∈𝔸_{(0)}^{1}} K_{n}(F (x)) → 0

is split exact.

100.D. Specialization. Let L be a field and let v be a discrete valuation on L with residue field F. If π ∈ L^{×} is a prime element, i.e., v(π) = 1, we define the specialization homomorphism

s_{π} : K_{∗}(L) → K_{∗}(F)

by the formula s_{π}(u) = ∂({−π} · u). We have

s_{π}({a_{1}, a_{2}, …, a_{n}}) = {b̄_{1}, b̄_{2}, …, b̄_{n}},

where b_{i} = a_{i}/π^{v(a_{i})}.

Example 100.6. Consider the discrete valuation v of the field of rational functions F (t) given by the irreducible polynomial t. For every u ∈ K_{∗}(F), we have s_{t}(u_{F (t)}) = u. In particular, the homomorphism K_{∗}(F) → K_{∗}(F (t)) is split injective as stated in Fact 100.5.

100.E. Norm homomorphism. Let L/F be a finite field extension. The standard norm homomorphism L^{×} → F^{×} can be viewed as a homomorphism K_{1}(L) → K_{1}(F). In fact, there exists the norm homomorphism

c_{L/F} : K_{n}(L) → K_{n}(F)

for every n ≥ 0 defined as follows.

Suppose first that the field extension L/F is simple, i.e., L is generated by one element over F. We identify L with the residue field F (y) of a closed point y ∈ 𝔸_{F}^{1} . Let α ∈ K_{n}(L) = K_{n}(F (y)). By Milnor’s Theorem 100.5, there is β ∈ K_{n+1}(F (𝔸_{F}^{1} )) satisfying

∂_{x}(β) = \{ \begin{matrix}α & if x = y, \\ 0 & otherwise. \end{matrix}

Let v be the discrete valuation of the field F (ℙ_{F}^{1} ) = F (𝔸_{F}^{1} ) associated with the infinite point of the projective line ℙ_{F}^{1} . We set c_{L/F} (α) = ∂_{v}(β).

In the general case, we choose a sequence of simple field extensions

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = L

and set

c_{L/F} = c_{F_{1}/F_{0}} ∘ c_{F_{2}/F_{1}} ∘ ⋯ ∘ c_{F_{n}/F_{n−1}}.

It turns out that the norm map c_{L/F} is well-defined, i.e., it does not depend on the choice of the sequence of simple field extensions and the identifications with residue fields of closed points of the affine line (cf. [43, Ch. IX, §3], [47, Ch. 7, §3]).

The following theorem is the direct consequence of the definition of the norm map and Milnor’s Theorem 100.5.

Theorem 100.7. For every field F, the sequence

0 → K_{n+1}(F) ⟶^{r_{F (t)/F}} K_{n+1}(F (t)) ⟶^{(∂_{x})} ∐_{x∈ℙ_{(0)}^{1}} K_{n}(F (x)) ⟶^{c} K_{n}(F) → 0

is exact where c is the direct sum of the corestriction homomorphisms c_{F (x)/F}.

Fact 100.8. (1) (Transitivity) Let L/F and E/L be finite field extensions. Then c_{E/F} = c_{L/F} ∘ c_{E/L}.

(2) The norm map c_{L/F} : K_{0}(L) → K_{0}(F) is multiplication by [L : F] on ℤ. The norm map c_{L/F} : K_{1}(L) → K_{1}(F) is the classical norm L^{×} → F^{×}.

(3) (Projection Formula) Let L/F be a finite field extension. Then for every α ∈ K_{∗}F and β ∈ K_{∗}(L) we have

c_{L/F} (r_{L/F} (α) · β) = α · c_{L/F} (β),

i.e., if we view K_{∗}(L) as a K_{∗}(F)-module via r_{L/F}, then c_{L/F} is a homomorphism of K_{∗}(F)-modules. In particular, the composition c_{L/F} ∘ r_{L/F} is multiplication by [L : F].

(4) Let L/F be a finite field extension and let v be a discrete valuation on F. Let v_{1}, v_{2}, …, v_{s} be all the extensions of v to L. Then the following diagram is commutative:

K_{n+1}(L) ⟶^{(∂_{v_{i}})} ∐_{i=1}^{s} K_{n}(L(v_{i}))
c_{L/F} ↓  ↓^{∑ c_{L(v_{i})/F (v)}}
K_{n+1}(F) ⟶^{∂_{v}} K_{n}(F (v)).

(5) Let L/F be a finite and E/F an arbitrary field extension. Let P_{1}, P_{2}, …, P_{k} be all the prime (maximal) ideals of the ring R = L ⊗_{F} E. For every i ∈ [1, k], let R_{i} denote the residue field R/P_{i} and let l_{i} be the length of the localization ring R_{P_{i}}. Then the following diagram is commutative:

K_{n}(L) ⟶^{(r_{R_{i}/L})} ∐_{i=1}^{k} K_{n}(R_{i})
c_{L/F} ↓  ↓^{∑ l_{i}·c_{R_{i}/E}}
K_{n}(F) ⟶^{r_{E/F}} K_{n}(E).

We now turn to fields of positive characteristic.

Fact 100.9 (Cf. [60, Th. A]). Let F be a field of characteristic p > 0. Then the p-torsion part of K_{∗}(F) is trivial.

Fact 100.10 (Cf. [60, Cor. 6.5]). Let F be a field of characteristic p > 0. Then the natural homomorphism

K_{n}(F)/pK_{n}(F) → H^{0}(F, K_{n}(F_{sep})/pK_{n}(F_{sep}))

is an isomorphism.

Now consider the case of purely inseparable quadratic extensions.

Lemma 100.11. Let L/F be a purely inseparable quadratic field extension. Then the composition r_{L/F} ∘ c_{L/F} on K_{n}(L) is the multiplication by 2.

Proof. The statement is obvious if n = 1. The general case follows from Proposition 100.2 and Fact 100.8(3). □

Write k_{n}(E) := K_{n}(E)/2K_{n}(E) for a field E.

Proposition 100.12. Let L/F be a purely inseparable quadratic field extension. Then the sequence

k_{n}(F) ⟶^{r_{L/F}} k_{n}(L) ⟶^{c_{L/F}} k_{n}(F) ⟶^{r_{L/F}} k_{n}(L)

is exact.

Proof. Let α ∈ K_{n}(F) satisfy α_{L} = 2β for some β ∈ K_{n}(L). By Proposition 100.8,

2α = c_{L/F} (α_{L}) = c_{L/F} (2β) = 2c_{L/F} (β),

hence α = c_{L/F} (β) in view of Fact 100.9.

Let β ∈ K_{n}(L) satisfy c_{L/F} (β) = 2α for some α ∈ K_{n}(F). It follows from Lemma 100.11 that

2β = c_{L/F} (β)_{L} = 2α_{L},

hence β = α_{L}, again by Fact 100.9. □

101. The cohomology groups H^{n,i}(F, ℤ/mℤ)

Let F be a field. For all n, m, i ∈ ℤ with m > 0, we define the group H^{n,i}(F, ℤ/mℤ) as follows (cf. [46]): If m is not divisible by char F, we set

H^{n,i}(F, ℤ/mℤ) = H^{n}(F, μ_{m}^{⊗i}),

where μ_{m}^{⊗i} is the ith tensor power of μ_{m} if i ≥ 0 and μ_{m}^{⊗i} = Hom(μ_{m}^{⊗−i}, ℤ/mℤ) if i < 0. If char F = p > 0 and m is a power of p, we set

H^{n,i}(F, ℤ/mℤ) = \{ \begin{matrix}K_{i}(F)/mK_{i}(F) & if n = i, \\ H^{1}(F, K_{i}(F_{sep})/mK_{i}(F_{sep})) & if n = i + 1, \\ 0 & otherwise. \end{matrix}

In the general case, write m = m_{1}m_{2}, where m_{1} is not divisible by char F and m_{2} is a power of char F if char F > 0, and set

H^{n,i}(F, ℤ/mℤ) = H^{n,i}(F, ℤ/m_{1}ℤ) ⊕ H^{n,i}(F, ℤ/m_{2}ℤ).

Note that if char F does not divide m and μ_{m} ⊂ F^{×}, we have a natural isomorphism

H^{n,i}(F, ℤ/mℤ) ≃ H^{n,0}(F, ℤ/mℤ) ⊗ μ_{m}^{⊗i}.

In particular, the groups H^{n,i}(F, ℤ/mℤ) and H^{n,0}(F, ℤ/mℤ) are (noncanonically) isomorphic.

Example 101.1. For an arbitrary field F, we have canonical isomorphisms (1) H^{0,0}(F, ℤ/mℤ) ≃ ℤ/mℤ,

(2) H^{1,1}(F, ℤ/mℤ) ≃ F^{×}/F^{×m},

(3) H^{1,0}(F, ℤ/mℤ) ≃ Hom_{c}(Γ_{F} , ℤ/mℤ), H^{1,0}(F, ℤ/2ℤ) ≅ Ét_{2}(F),

(4) H^{2,1}(F, ℤ/mℤ) ≃ Br_{m}(F).

If L/F is a field extension, there is the restriction homomorphism

r_{L/F} : H^{n,i}(F, ℤ/mℤ) → H^{n,i}(L, ℤ/mℤ).

If L is a finite over F, we define the corestriction homomorphism

c_{L/F} : H^{n,i}(L, ℤ/mℤ) → H^{n,i}(F, ℤ/mℤ)

as follows: It is sufficient to consider the following two cases.

101. THE COHOMOLOGY GROUPS H^{n,i}(F, ℤ/mℤ) 403

(1) If L/F is separable, then c_{L/F} is the corestriction homomorphism in Galois cohomology.

(2) If L/F is purely inseparable, then Γ_{L} = Γ_{F}, we have [L_{sep} : F_{sep}] = [L : F], and c_{L/F} is induced by the corestriction homomorphism K_{∗}(L_{sep}) → K_{∗}(F_{sep}).

Example 101.2. Let L/F be a finite field extension. By Example 99.6, the map

c_{L/F} : L^{×}/L^{×m} = H^{1,1}(L, ℤ/mℤ) → H^{1,1}(F, ℤ/mℤ) = F^{×}/F^{×m}

is induced by the norm map N_{L/F} : L^{×} → F^{×}. If char F = p > 0, it follows from Example 99.1(2) that the map

c_{L/F} : L/℘(L) = H^{1,0}(L, ℤ/pℤ) → H^{1,0}(F, ℤ/pℤ) = F/℘(F)

is induced by the trace map Tr_{L/F} : L → F.

Let l, m ∈ Z. If char F does not divide l and m, we have a natural exact sequence of Galois modules

1 → μ_{l}^{⊗i} → μ_{lm}^{⊗i} → μ_{m}^{⊗i} → 1

for every i. If l and m are powers of char F > 0, then by Fact 100.9, the sequence of Galois modules

0 → K_{n}(F_{sep})/lK_{n}(F_{sep}) → K_{n}(F_{sep})/lmK_{n}(F_{sep}) → K_{n}(F_{sep})/mK_{n}(F_{sep}) → 0

is exact. Taking the long exact sequences of Galois cohomology groups yields the following proposition.

Proposition 101.3. For any l, m, n, i ∈ ℤ with l, m > 0, there is a natural long exact sequence

⋯ → H^{n,i}(F, ℤ/lℤ) → H^{n,i}(F, ℤ/lmℤ)

→ H^{n,i}(F, ℤ/mℤ) → H^{n+1,i}(F, ℤ/lℤ) → · · · .

The cup-product in Galois cohomology and the product in the Milnor ring induce a structure of the graded ring on the graded abelian group

H^{∗,∗}(F, ℤ/mℤ) = ∐_{i,j∈ℤ} H^{i,j}(F, ℤ/mℤ)

for every m ∈ ℤ. The product in this ring will be denoted by ∪.

101.A. Norm residue homomorphism. Let symbol (a)_{m} denote the element in H^{1,1}(F, ℤ/mℤ) corresponding to a ∈ F^{×} under the isomorphism in Example 101.1(2).

Lemma 101.4 (Steinberg Relation). Let a, b ∈ F^{×} satisfy a + b = 1. Then (a)_{m} ∪ (b)_{m} = 0 in H^{2,2}(F, ℤ/mℤ).

Proof. We may assume that char F does not divide m. Let K = F [t]/(t^{m} −a) and let α ∈ K^{×} be the class of t. We have a = α^{m} and N_{K/F} (1 − α) = b, hence 1 − α ∈ K^{×}. It follows from the projection formula and Example 99.6 that

(a)_{m} ∪ (b)_{m} = c_{K/F} (r_{K/F} (a)_{m} ∪ (1 − α)_{m}) = 0

since r_{K/F} (a)_{m} = m(α)_{m} = 0 in H^{1,1}(K, ℤ/mℤ). □

It follows from Lemma 101.4 that for every n, m ∈ ℤ there is a unique norm residue homomorphism

(101.5)  h_{F}^{n,m} : K_{n}(F)/mK_{n}(F) → H^{n,n}(F, ℤ/mℤ)

taking the class of a symbol {a_{1}, a_{2}, …, a_{n}} to the cup-product (a_{1})_{m} ∪ (a_{2})_{m} ∪ ⋯ ∪ (a_{n})_{m}.

The norm residue homomorphism allows us to view H^{∗,∗}(F, ℤ/mℤ) as a module over the Milnor ring K_{∗}(F).

By Example 99.1, the map h_{F}^{n,m} is an isomorphism for n = 0 and 1. Bloch and Kato conjectured that h_{F}^{n,m} is always an isomorphism.

For every l, m ∈ ℤ, we have a commutative diagram

K_{n}(F)/lmK_{n}(F) ⟶ K_{n}(F)/mK_{n}(F)
h_{F}^{n,lm}↓  ↓^{h_{F}^{n,m}}
H^{n,n}(F, ℤ/lmℤ) ⟶ H^{n,n}(F, ℤ/mℤ)

with top map the natural surjective homomorphism.

The following important result was originally conjectured by Milnor in [106] and was proven in [136] by Voevodsky.

Fact 101.6. Let F be a field of characteristic not 2. If m is a power of 2, then the norm residue homomorphism h_{F}^{n,m} is an isomorphism.

Proposition 101.3 and commutativity of the diagram above yield

Corollary 101.7. Let l and m be powers of 2. Then the natural homomorphism H^{n,n}(F, ℤ/lmℤ) → H^{n,n}(F, ℤ/mℤ) is surjective and the sequence

0 → H^{n+1,n}(F, ℤ/lℤ) → H^{n+1,n}(F, ℤ/lmℤ) → H^{n+1,n}(F, ℤ/mℤ)

is exact for any n.

Now consider the case m = 2. We shall write h_{F}^{n} for h_{F}^{n,2} and H^{n}(F) for H^{n,n}(F, ℤ/2ℤ).

The norm residue homomorphisms commute with field extension homomorphisms. They also commute with residue and corestriction homomorphisms as the following two propositions show.

Proposition 101.8. Let L be a field with a discrete valuation v and residue field F of characteristic different from 2. Then the diagram

k_{n+1}(L) ⟶^{∂_{v}} k_{n}(F)
h_{L}^{n+1}↓  ↓^{h_{F}^{n}}
H^{n+1}(L) ⟶^{∂_{v}} H^{n}(F)

is commutative.

Proof. Fact 99.11(1) shows that the diagram is commutative when n = 0. The general case follows from Fact 99.11(2) as the group k_{n+1}(L) is generated by symbols {a_{0}, a_{1}, …, a_{n}} with v(a_{1}) = ⋯ = v(a_{n}) = 0. □

Proposition 101.9. Let L/F be a finite field extension. Then the diagram

k_{n}(L) ⟶^{c_{L/F}} k_{n}(F)
h_{L}^{n}↓  ↓^{h_{F}^{n}}
H^{n}(L) ⟶^{c_{L/F}} H^{n}(F)

is commutative.

Proof. We may assume that L/F is a simple field extension. The statement follows from the definition of the norm map for the Milnor K-groups, Fact 99.12, and Proposition 101.8. □

Proposition 101.10. Let F be a field of characteristic different from 2 and let L = F (\sqrt{a})/F be a quadratic extension with a ∈ F^{×}. Then the following infinite sequence

… → k_{n−1}(F) ⟶^{{a}} k_{n}(F) ⟶^{r_{L/F}} k_{n}(L) ⟶^{c_{L/F}} k_{n}(F) ⟶^{{a}} k_{n+1}(F) → …

is exact.

Proof. It follows from Proposition 101.9 that the diagram

k_{n−1}(F) ⟶^{{a}} k_{n}(F) ⟶^{r_{L/F}} k_{∗}(L) ⟶^{c_{L/F}} k_{n}(F) ⟶^{{a}} k_{n+1}(F)
h_{F}^{n−1}↓  ^{h_{F}^{n}} ↓  ^{h_{L}^{n}}↓  ^{h_{F}^{n}} ↓  ↓^{h_{F}^{n+1}}
H^{n−1}(F) ⟶^{{a}} H^{n}(F) ⟶^{r_{L/F}} H^{∗}(L) ⟶^{c_{L/F}} H^{n}(F) ⟶^{{a}} H^{n+1}(F)

is commutative. By Fact 101.6, the vertical homomorphisms are isomorphisms. By Theorem 99.13, the bottom sequence is exact. The result follows. □

Now consider the case char F = 2. The product in the Milnor ring and the cup-product in Galois cohomology yield a pairing

K_{∗}(F) ⊗ H^{∗}(F) → H^{∗}(F)

making H^{∗}(F) a module over K_{∗}(F).

Example 101.11. By Example 99.5, we have \{a\} · [F_{ab}] = \left[{a,b \atop F}\right] in Br_{2}(F) for all a ∈ F^{×} and b ∈ F.

Proposition 101.12. Let F be a field of characteristic 2 and let L/F be a separable quadratic field extension. Then the following sequence

0 → k_{n}(F) ⟶^{r_{L/F}} k_{n}(L) ⟶^{c_{L/F}} k_{n}(F)

⟶^{·[L]} H^{n+1}(F) ⟶^{r_{L/F}} H^{n+1}(L) ⟶^{c_{L/F}} H^{n+1}(F) → 0

is exact where the middle map is multiplication by the class of L in H^{1}(F).

Proof. We shall show that the sequence in question coincides with the exact sequence in Theorem 99.13 for the quadratic field extension L/F and the Galois module k_{n}(F_{sep}) over F. Indeed, by Fact 100.10, we have H^{0}(E, k_{n}(E_{sep})) ≃ k_{n}(E) and by definition, H^{1}(E, k_{n}(E_{sep})) ≃ H^{n+1}(E) for every field E. Note that H^{2}(F, k_{n}(F_{sep})) = 0 by Example 99.1(2). The connecting homomorphism in the sequence in Theorem 99.13 is multiplication by the class of L in H^{1}(F). □

Now let F be a field of characteristic different from 2. The connecting homomorphism

b_{n} : H^{n}(F) → H^{n+1}(F)

with respect to the short exact sequence

(101.13)  0 → ℤ/2ℤ → ℤ/4ℤ → ℤ/2ℤ → 0

is called the Bockstein map.

Proposition 101.14. The Bockstein map is trivial if n is even and coincides with multiplication by (−1) if n is odd.

Proof. If n is even or −1 ∈ F^{×2}, then μ_{4}^{⊗n} ≃ ℤ/4ℤ and the statement follows from Corollary 101.7.

Suppose that n is odd and −1 ∉ F^{×2}. In this case μ_{4}^{⊗n} ≃ μ_{4}. Consider the field K = F (\sqrt{−1}). By Theorem 99.13, the connecting homomorphism H^{n}(F) → H^{n+1}(F) with respect to the exact sequence (99.9) is the cup-product with (−1). The classes of the sequences (101.13) and (99.9) differ in Ext_{Γ}^{1}(ℤ/2ℤ, ℤ/2ℤ) by the class of the sequence

0 → ℤ/2ℤ → μ_{4}^{⊗n} → ℤ/2ℤ → 0.

By Corollary 101.7, the connecting homomorphism H^{n}(F) → H^{n+1}(F) with respect to this exact sequence is trivial. It follows that b_{n} is the cup-product with (−1). □

101.B. Cohomological dimension and p-special fields. Let p be a prime integer. A field F is called p-special if the degree of every finite field extension of F is a power of p.

The following property of p-special fields is very useful.

Proposition 101.15. Let F be a p-special field and let L/F be a finite field extension. Then there is a tower of field extensions

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n−1} ⊂ F_{n} = L

satisfying [F_{i+1} : F_{i}] = p for all i ∈ [0, n − 1].

Proof. The result is clear if L/F is purely inseparable. So we may assume that L/F is a separable extension. Let E/F be a normal closure of L/F. Set G = Gal(E/F) and H = Gal(E/L). As G is a p-group, there is a sequence of subgroups

G = H_{0} ⊃ H_{1} ⊃ ⋯ ⊃ H_{n−1} ⊃ H_{n} = H

with the property [H_{i} : H_{i+1}] = p for all i ∈ [0, n − 1]. Then the fields F_{i} = L^{H_{i}} satisfy the required property. □

Proposition 101.16. For every prime integer p and field F, there is a field extension L/F satisfying

(1) L is p-special.

(2) The degree of every finite sub-extension of L/F is not divisible by p.

Proof. If char F = q > 0 and different from p, we set F′ := ⋃ F^{q^{−n}} (in a fixed algebraic closure of F), otherwise F′ := F. Let Γ be the Galois group of F′_{sep}/F′ and let Δ ⊂ Γ be a Sylow p-subgroup. The field L = (F′_{sep})^{Δ} satisfies the required conditions. □

We call the field L in Proposition 101.16 a p-special closure of F.

Let F be a field and let p be a prime integer. The cohomological p-dimension of F, denoted cd_{p}(F), is the smallest integer such that for every n > cd_{p}(F) and every finite field extension L/F we have H^{n,n−1}(L, ℤ/pℤ) = 0.

Example 101.17. (1) cd_{p}(F) = 0 if and only if F has no separable finite field extensions of degree a power of p.

(2) cd_{p}(F) ≤ 1 if and only if Br_{p}(L) = 0 for all finite field extensions L/F.

(3) If F is p-special, then cd_{p}(F) < n if and only if H^{n,n−1}(F, ℤ/pℤ) = 0.

102. Length and Herbrand index

102.A. Length. Let A be a commutative ring and let M be an A-module of finite length. The length of M is denoted by l_{A}(M). The ring A is artinian if the A-module M = A is of finite length. We write l(A) for l_{A}(A).

Lemma 102.1. Let C be a flat B-algebra where B and C are commutative local artinian rings. Then for every finitely generated B-module M, we have

l_{C}(M ⊗_{B} C) = l(C/QC) · l_{B}(M),

where Q is the maximal ideal of B.

Proof. Since C is flat over B, both sides of the equality are additive in M. Thus, we may assume that M is a simple B-module, i.e., M = B/Q. We have M ⊗_{B} C ≃ C/QC and the equality follows. □

Setting M = B we obtain

Corollary 102.2. In the conditions of Lemma 102.1, one has l(C) = l(C/QC) · l(B).

Lemma 102.3. Let B be a commutative A-algebra and let M be a B-module of finite length over A. Then

l_{A}(M) = ∑ l_{B_{Q}}(M_{Q}) · l_{A}(B/Q),

where the sum is taken over all maximal ideals Q ⊂ B.

Proof. Both sides are additive in M, so we may assume that M = B/Q, where Q is a maximal ideal of B. The result follows. □

102.B. Herbrand index. Let M be a module over a commutative ring A and let a ∈ A. Suppose that the modules M/aM and let _{a}M := Ker(M ⟶^{a} M) have finite length. The integer

h(a, M) = l_{A}(M/aM) − l_{A}(_{a}M)

is called the Herbrand index of M relative to a.

We collect simple properties of the Herbrand index in the following lemma.

Lemma 102.4. (1) Let 0 → M′ → M → M′′ → 0 be an exact sequence of Amodules. Then h(a, M) = h(a, M′) + h(a, M′′).

(2) If M has finite length, then h(a, M) = 0.

Lemma 102.5. Let S be a 1-dimensional Noetherian local ring and let P_{1}, …, P_{m} be all the minimal prime ideals of S. Let M be a finitely generated S-module and s ∈ S not belonging to any of P_{i}. Then

h(s, M) = ∑_{i=1}^{m} l_{S_{P_{i}}} (M_{P_{i}}) · l(S/(P_{i} + sS)).

Proof. Since s ∉ P_{i}, the coset of s in S/P_{i} is not a zero divisor. Hence

l_{S}(S/(P_{i} + sS)) = h(s, S/P_{i}).

Both sides of the equality are additive in M. Since M has a filtration with factors S/P, where P is a prime ideal of S, we may assume that M = S/P. If P is maximal, then M_{P_{i}} = 0 and h(s, M) = 0 since M is simple. If P = P_{i} for some i, then l_{S_{P_{j}}} (M) = 1 if i = j and zero otherwise. The equality holds in this case too. □

103. Places

Let K be a field. A valuation ring R of K is a subring R ⊂ K such that for any x ∈ K \ R, we have x^{−1} ∈ R. A valuation ring is a local domain. A trivial example of a valuation ring is the field K itself.

Given two fields K and L, a place π : K ⇀ L is a local ring homomorphism f : R → L of a valuation ring R ⊂ K. We say that the place π is defined on R. An embedding of fields is a trivial example of a place defined everywhere. A place K ⇀ L is called surjective if f is surjective.

If K and L are extensions of a field F, we say that a place K ⇀ L is an F -place if π is defined and the identity on F.

Let K ⇀ L and L ⇀ E be two places, given by ring homomorphisms f : R → L and g : S → E respectively, where R ⊂ K and S ⊂ L are valuation rings. Then

the ring T = f^{−1}(S) is a valuation ring of K and the composition T ⟶^{f|_{T}} S ⟶^{g} E defines the composition place K ⇀ E. In particular, any place L ⇀ E can be restricted to any subfield K ⊂ L.

A composition of two F -places is an F -place. Every place is a composition of a surjective place and a field embedding.

A place K ⇀ L is said to be geometric if it is a composition of (finitely many) places each defined on a discrete valuation ring. An embedding of fields will also be viewed as a geometric place.

Let Y be a complete variety over F and let π : F (Y) ⇀ L be an F -place. The valuation ring R of the place dominates a unique point y ∈ Y, i.e., O_{Y,y} ⊂ R and the maximal ideal of O_{Y,y} is contained in the maximal ideal M of R. The induced homomorphism of fields F (y) → R/M → L over F gives rise to an L-point of Y, i.e., to a morphism f : Spec(L) → Y with image {y}. We say that y is the center of π and f is induced by π.

Let X be a regular variety over F and let f : Spec(L) → X be a morphism over F. Choose a regular system of parameters a_{1}, a_{2}, …, a_{n} in the local ring R = O_{X,x}, where {x} is the image of f. Let M_{i} be the ideal of R generated by a_{1}, …, a_{i} and set R_{i} = R/M_{i}, P_{i} = M_{i+1}/M_{i}. Denote by F_{i} the quotient field of R_{i}, in particular, F_{0} = F (X) and F_{n} = F (x). The localization ring (R_{i})_{P_{i}} is a discrete valuation ring with quotient field F_{i} and residue field F_{i+1} therefore, it determines a place F_{i} ⇀ F_{i+1}. The composition of places

F (X) = F_{0} ⇀ F_{1} ⇀ … ⇀ F_{n} = F (x) ↪ L

is a geometric place constructed (noncanonically) out of the point f ∈ X(L).

Lemma 103.1. Let K be an arbitrary field, let K′/K be an odd degree field extension, and let L/K be an arbitrary field extension. Then there exists a field L′, containing K′ and L, such that the extension L′/L is of odd degree.

Proof. We may assume that K′/K is a simple extension, i.e., K′ is generated over K by one element. Let f(t) ∈ F [t] be its minimal polynomial. Since the degree of f is odd, there exists an irreducible divisor g ∈ L[t] of f over L with odd deg(g). We set L′ = L[t]/(g). □

Lemma 103.2. Let K be a field extension of F of finite transcendence degree over F, let K ⇀ L be a geometric F -place and let K′ be a finite field extension of K of odd degree. Then there exists an odd degree field extension L′/L such that the place K ⇀ L extends to a place K′ ⇀ L′.

Proof. By Lemma 103.1, it suffices to consider the case of a surjective place K ⇀ L given by a discrete valuation ring R. It also suffices to consider only two cases: (1) K′/K is purely inseparable and (2) K′/K is separable.

In the first case, the degree [K′ : K] is a power of an odd prime p. Let R′ be an arbitrary valuation ring of K′ lying over R, i.e., such that R′ ∩ K = R and with the embedding R → R′ local (such an R′ exists in the case of an arbitrary field extension K′/K by [140, Ch. VI, Th. 5′]). We have a surjective place K′ ⇀ L′, where L′ is the residue field of R′. We claim that L′ is purely inseparable over L (and therefore [L′ : L], being a power of p, is odd). Indeed if l ∈ L′, choose a preimage k ∈ R′ of l. Then k^{p^{n}} ∈ K for some n, hence l^{p^{n}} ∈ L, i.e., L′/L is a purely inseparable extension.

In the second case, consider all the valuation rings R_{1}, …, R_{r} of K′ lying over R (the number of such valuation rings is finite by [140, Ch. VI, Th. 12, Cor. 4]). The residue field of each R_{i} is a finite extension of L. Moreover, ∑_{i=1}^{r} e_{i}n_{i} = [K′ : K] by [140, Ch. VI, Th. 20 and p. 63], where n_{i} is the degree over L of the residue field of R_{i}, and e_{i} is the ramification index of R_{i} over R (cf. [140, Def. on pp. 52–53]). It follows that at least one of the n_{i} is odd. □

104. Cones and vector bundles

The word “scheme” in the next two sections means a separated scheme of finite type over a field.

104.A. Definition of a cone. Let X be a scheme over a field F and let S^{•} = S^{0} ⊕ S^{1} ⊕ S^{2} ⊕ … be a sheaf of graded O_{X}-algebras. We assume that

(1) The natural morphism O_{X} → S^{0} is an isomorphism.

(2) The O_{X}-module S^{1} is coherent.

(3) The sheaf of O_{X}-algebras S^{•} is generated by S^{1}.

The cone of S^{•} is the scheme C = Spec(S^{•}) over X and ℙ(C) = Proj(S^{•}) is called the projective cone of C. Recall that Proj(S^{•}) has a covering by the principal open subschemes D(s) := Spec(S_{(s)}) over all s ∈ S^{1}, where S_{(s)} is the subring of degree 0 elements in the localization S_{s}.

We have natural morphisms C → X and ℙ(C) → X. The canonical homomorphism S^{•} → S^{0} of O_{X}-algebras induces the zero section X → C.

If C and C′ are cones over X, then C ×_{X} C′ has a natural structure of a cone over X. We denote it by C ⊕ C′.

Example 104.1. A coherent O_{X}-module P defines the cone C(P) = Spec(S^{•}(P)) over X, where S^{•} stands for the symmetric algebra. If the sheaf P is locally free, the cone E := C(P) is called the vector bundle over X with the dual sheaf of sections P^{∨} := Hom_{O_{X}} (P, O_{X}). The projective cone ℙ(E) is called the projective bundle of E. The assignment P ↦ C(P^{∨}) gives rise to an equivalence between the category of locally free coherent O_{X}-modules and the category of vector bundles over X. In particular, such operations over the locally free O_{X}-modules as the tensor product, symmetric power, dual sheaf etc., and the notion of an exact sequence translate to the category of vector bundles. We write K_{0}(X) for the Grothendieck group of the category of vector bundles over X. The group K_{0}(X) is the abelian group given by generators, the isomorphism classes [E] of vector bundles E over X and relations [E] = [E′] + [E′′] for every exact sequence 0 → E′ → E → E′′ → 0 of vector bundles over X.

Example 104.2. The trivial line bundle X × 𝔸^{1} → X will be denoted by 𝟙.

Example 104.3. Let f : Y → X be a closed embedding and let I ⊂ O_{X} be the sheaf of ideals of the image of f in X. The cone

C_{f} = Spec(O_{X}/I ⊕ I/I^{2} ⊕ I^{2}/I^{3} ⊕ …)

over Y is called the normal cone of Y in X. If X is a scheme of pure dimension d, then C_{f} is also a scheme of pure dimension d [45, B.6.6].

Example 104.4. If f : X → C is the zero section of a cone C, then C_{f} = C.

Example 104.5. The cone T_{X} := C_{f} of the diagonal embedding f : X → X × X is called the tangent cone of X. If X is a scheme of pure dimension d, then the tangent cone T_{X} is a scheme of pure dimension 2d (cf. Example 104.3).

Let U and V be vector spaces over a field F and let

U = U_{0} ⊃ U_{1} ⊃ U_{2} ⊃ … and V = V_{0} ⊃ V_{1} ⊃ V_{2} ⊃ …

be two filtrations by subspaces. Consider the filtration on U ⊗ V defined by

(U ⊗ V)_{k} = ∑_{i+j=k} U_{i} ⊗ V_{j}.

The following lemma can be proven by a suitable choice of bases of U and V.

Lemma 104.6. The canonical linear map

∐_{i+j=k}(U_{i}/U_{i+1}) ⊗ (V_{j}/V_{j+1}) → (U ⊗ V)_{k}/(U ⊗ V)_{k+1}

is an isomorphism for every k ≥ 0.

Proposition 104.7. Let f : Y → X and g : S → T be closed embeddings. Then there is a canonical isomorphism C_{f} × C_{g} ≃ C_{f×g}.

Proof. We may assume that X = Spec(A), Y = Spec(A/I), and T = Spec(B), S = Spec(B/J), where I ⊂ A and J ⊂ B are ideals. Then X × T = Spec(A ⊗ B) and Y × S = Spec(A ⊗ B)/K, where K = I ⊗ B + A ⊗ J.

Consider the vector spaces U_{i} = I^{i} and V_{j} = J^{j}. We have (U ⊗ V)_{k} = K^{k}. By Lemma 104.6,

C_{f} × C_{g} = Spec ( ∐_{i≥0} I^{i}/I^{i+1} ⊗ ∐_{j≥0} J^{j}/J^{j+1}) ≃ Spec ( ∐_{k≥0} K^{k}/K^{k+1}) = C_{f×g}. □

Corollary 104.8. If X and Y are two schemes, then T_{X×Y} = T_{X} × T_{Y}.

104.B. Regular closed embeddings. Let A be a commutative ring. A sequence 𝔞 = (a_{1}, a_{2}, …, a_{d}) of elements of A is called regular if the ideal generated by a_{1}, …, a_{d} is different from A and the coset of a_{i} is not a zero divisor in the factor ring A/(a_{1}A + ⋯ + a_{i−1}A) for all i ∈ [1, d]. We write l(𝔞) = d.

Let Y be a scheme and let d : Y → ℤ be a locally constant function. A closed embedding f : Y → X is called regular of codimension d if for every point y ∈ Y there is an affine neighborhood U ⊂ X of f(y) such that the ideal of f(Y) ∩ U in F [U] is generated by a regular sequence of length d(y).

Let f : Y → X be a closed embedding and let I be the sheaf of ideals of Y in O_{X}. The embedding of I/I^{2} into ∐_{k≥0} I^{k}/I^{k+1} induces a surjective O_{Y} -algebra homomorphism S^{•}(I/I^{2}) → ∐_{k≥0} I^{k}/I^{k+1} and therefore a closed embedding of cones φ_{f} : C_{f} → C(I/I^{2}) over Y.

Proposition 104.9. (Cf. [48, Cor. 16.9.4, Cor. 16.9.11].) A closed embedding f : Y → X is regular of codimension d if and only if the O_{Y} -module I/I^{2} is locally free of rank d and the natural morphism φ_{f} : C_{f} → C(I/I^{2}) is an isomorphism.

Corollary 104.10. Let f : Y → X be a regular closed embedding of codimension d and let I be the sheaf of ideals of Y in O_{X}. Then the normal cone C_{f} is a vector bundle over Y of rank d with the sheaf of sections naturally isomorphic to (I/I^{2})^{∨}.

We shall write N_{f} for the normal cone C_{f} of a regular closed embedding f and call N_{f} the normal bundle of f.

Proposition 104.11. Let f : Y → X be a closed embedding and let g : X′ → X be a faithfully flat morphism. Then f is a regular closed embedding if and only if the closed embedding f′ : Y′ = Y ×_{X} X′ → X′ is regular.

Proof. Let I be the sheaf of ideals of Y in O_{X}. Then I′ = g^{∗}(I) is the sheaf of ideals of Y′ in O_{X′}. Moreover

g^{∗}(I^{k}/I^{k+1}) = I′^{k}/I′^{k+1}, C_{f} ×_{Y} Y′ = C_{f′}, C(I/I^{2}) ×_{Y} Y′ = C(I′/I′^{2}),

and φ_{f} ×_{Y} 1_{Y′} = φ_{f′}. By faithfully flat descent, I/I^{2} is locally free and φ_{f} is an isomorphism if and only if I′/I′^{2} is locally free and φ_{f′} is an isomorphism. The statement follows by Proposition 104.9. □

Proposition 104.12 (Cf. [48, Cor. 17.12.3]). Let g : X → Y be a smooth morphism of relative dimension d and let f : Y → X be a section of g, i.e., g ∘ f = 1_{Y}. Then f is a regular closed embedding of codimension d and N_{f} = f^{∗}T_{g}, where T_{g} := Ker(T_{X} → g^{∗}T_{Y}) is the relative tangent bundle of g over X.

Corollary 104.13. Let X be a smooth scheme. Then the diagonal embedding X → X × X is regular. In particular, the tangent cone T_{X} is a vector bundle over X.

Proof. The diagonal embedding is a section of any of the two projections X × X → X. □

If X is a smooth scheme, the vector bundle T_{X} is called the tangent bundle over X.

Corollary 104.14. Let f : X → Y be a morphism where Y is a smooth scheme of pure dimension d. Then the morphism h = (1_{X}, f) : X → X × Y is a regular closed embedding of codimension d with N_{h} = f^{∗}T_{Y}.

Proof. Applying Proposition 104.12 to the smooth projection p : X ×Y → X of relative dimension d, we have the closed embedding h is regular of codimension d. The tangent bundle T_{p} is equal to q^{∗}T_{Y}, where q : X × Y → Y is the other projection. Since q ∘ h = f, we have

N_{h} = h^{∗}T_{p} = h^{∗} ∘ q^{∗}T_{Y} = f^{∗}T_{Y} .  □

Proposition 104.15 (Cf. [48, Prop. 19.1.5]). Let g : Z → Y and f : Y → X be regular closed embeddings of codimension s and r respectively. Then f ∘ g is a regular closed embedding of codimension r + s and the natural sequence of normal bundles over Z,

0 → N_{g} → N_{f∘g} → g^{∗}N_{f} → 0,

is exact.

Proposition 104.16 (Cf. [48, Th. 17.12.1, Prop. 17.13.2]). A closed embedding f: Y → X of smooth schemes is regular and the natural sequence of vector bundles over Y,

0 → T_{Y} → f^{∗}T_{X} → N_{f} → 0,

is exact.

104.C. Canonical line bundle. Let C = Spec(S^{•}) be a cone over X. The cone Spec(S^{•}[t]) = C × 𝔸^{1} coincides with C ⊕ 𝟙. Let I ⊂ S^{•}[t] be the ideal generated by S^{1}. The closed subscheme of ℙ(C ⊕ 𝟙) defined by I is isomorphic to Proj(S^{0}[t]) = Spec(S^{0}) = X. Thus we get a canonical closed embedding (canonical section) of X into ℙ(C ⊕ 𝟙).

Set L_{c} := ℙ(C ⊕ 𝟙) \ X. The inclusion of S_{(s)}^{•} into S^{•}[t]_{(s)} for every s ∈ S^{1} induces a morphism L_{c} → ℙ(C).

Proposition 104.17. The morphism L_{c} → ℙ(C) has a canonical structure of a line bundle.

Proof. We have S^{•}[t]_{(s)} = S_{(s)}^{•}[\frac{t}{s}], hence the preimage of D(s) is isomorphic to D(s) × 𝔸^{1}. For any other element s′ ∈ S^{1} we have \frac{t}{s′} = \frac{s}{s′} \frac{t}{s}, i.e., the change of coordinate function is linear. □

The line bundle L_{c} → ℙ(C) is called the canonical line bundle over ℙ(C). A section of L_{c} over the open set D(s) is given by an S_{(s)}^{•}-algebra homomor-

phism S_{(s)}^{•}[\frac{t}{s}] → S_{(s)}^{•} that is uniquely determined by the image a_{s} of \frac{t}{s}. The element sa_{s} ∈ S_{s} of degree 1 agrees with s′a_{s′} on the intersection D(s) ∩ D(s′). Therefore the sheaf of sections of L_{c} coincides with O(1) := S̃^{•}(1) where S̃^{•} denotes the sheaf associated to S^{•} (cf. [50, Ch. II, §5]).

The scheme ℙ(C) can be viewed as a locally principal divisor of ℙ(C ⊕ 𝟙) given by t. The open complement ℙ(C ⊕ 𝟙) \ ℙ(C) is canonically isomorphic to C. The image of the canonical section X → ℙ(C ⊕𝟙) is contained in C (and in fact is equal to the image of the zero section of C), hence it does not intersect ℙ(C). Moreover, ℙ(C ⊕ 𝟙) \ (ℙ(C) ∪ X) is canonically isomorphic to C \ X.

If C is a cone over X, we write C^{∘} for C \ X where X is viewed as a closed subscheme of C via the zero section. We have shown that C^{∘} is canonically isomorphic to L_{c}^{∘}. Note that C is a cone over X and L_{c} is a cone (in fact, a line bundle) over ℙ(C).

For every s ∈ S^{1}, the localization S_{s} is the Laurent polynomial ring S_{(s)}[s, s^{−1}] over S_{(s)}. Hence the inclusion of S_{(s)} into S_{s} induces a flat morphism C^{∘} → ℙ(C) of relative dimension 1.

104.D. Tautological line bundle. Let C = Spec(S^{•}) be a cone over X. Consider the tensor product T^{•} = S^{•} ⊗_{S^{0}} S^{•} as a graded ring with respect to the second factor. We have

Proj(T^{•}) = C ×_{X} ℙ(C).

Let J be the ideal of T^{•} generated by x ⊗ y − y ⊗ x for all x, y ∈ S^{1} and set

L_{t} := Proj(T^{•}/J).

Thus L_{t} is a closed subscheme of C ×_{X} ℙ(C) and we have natural projections L_{t} → C and L_{t} → ℙ(C).

Proposition 104.18. The morphism L_{t} → ℙ(C) has a canonical structure of a line bundle.

Proof. Let s ∈ S^{1}. The preimage of D(s) in L_{t} coincides with

Spec(T_{(1⊗s)}^{•}/J_{(1⊗s)}),

where J_{(1⊗s)} = J_{1⊗s}∩T_{(1⊗s)}^{•}. The homomorphism T^{•} → S_{s}^{•}[t], where t is a variable, defined by x ⊗ y ↦ \frac{xy}{s^{n}} · t^{n} for any x ∈ S^{n} and y ∈ S^{•}, gives rise to an isomorphism between T_{(1⊗s)}^{•}/J_{(1⊗s)} and S_{(s)}^{•}[t]. Hence the preimage of D(s) is isomorphic to D(s) × 𝔸^{1}. □

The line bundle L_{t} → ℙ(C) is called the tautological line bundle over ℙ(C).

Example 104.19. If L is a line bundle over X, then ℙ(L) = X and L_{t} = L ×_{X} ℙ(L) = L.

Similar to the case of the canonical line bundle, a section of L_{t} over the open set D(s) is given by an element a_{s} ∈ S_{(s)}^{•} and the element a_{s}/s ∈ S_{s}^{•} of degree −1 agrees with a_{s′}/s′ on the intersection D(s) ∩ D(s′). Therefore the sheaf of sections of L_{t} coincides with S̃^{•}(−1) = O(−1). In particular, the tautological line bundle is dual to the canonical line bundle, L_{t} = L_{c}^{∨}.

The ideal I = S^{>0} in S^{•} defines the image of the zero section of C. The graded ring T^{•}/J is isomorphic to S^{•} ⊕ I ⊕ I^{2} ⊕ · · ·. Therefore, the canonical morphism L_{t} → C is the blowup of C along the image of the zero section of C. The exceptional divisor in L_{t} is the image of the zero section of L_{t}. Hence the induced morphism L_{t}^{∘} → C^{∘} is an isomorphism.

Example 104.20. Let F [ε] be the F -algebra of dual number over F. The tangent space T_{ℙ(V),L} of the point of the projective space ℙ(V) given by a line L ⊂ V coincides with the fiber over L of the map ℙ(V)(F [ε]) → ℙ(V)(F) induced by the ring homomorphism F [ε] → F, ε ↦ 0. For example, the F [ε]-submodule L ⊕ Lε of V [ε] := V ⊗ F [ε] represents the zero vector of the tangent space T_{ℙ(V),L}.

For a linear map h : L → V let W_{h} be the F [ε]-submodule of V [ε] generated by the elements v + h(v)ε, v ∈ L. Since W_{h}/εW_{h} ≃ L, we can view W_{h} as a point of T_{ℙ(V),L}. The map Hom_{F} (L, V) → T_{ℙ(V),L} given by h ↦ W_{h} yields an exact sequence of vector spaces

0 → Hom_{F} (L, L) → Hom_{F} (L, V) → T_{ℙ(V),L} → 0.

In other words,

T_{ℙ(V),L} = Hom_{F} (L, V/L).

Since the fiber of the tautological line bundle L_{t} over the point given by L coincides with L, we get an exact sequence of vector bundles over ℙ(V):

0 → Hom(L_{t}, L_{t}) → Hom(L_{t}, 𝟙 ⊗_{F} V) → T_{ℙ(V)} → 0.

The first term of the sequence is isomorphic to 𝟙 and the second term is isomorphic to L_{c} ⊗_{F} V ≃ (L_{c})^{⊕n}, where n = dim V. It follows that

[T_{ℙ(V)}] = n[L_{c}] − 1 ∈ K_{0}(ℙ(V)).

More generally, if E → X is a vector bundle, then there is an exact sequence of vector bundles over ℙ(E):

0 → 𝟙 → L_{c} ⊗ q^{∗}E → T_{q} → 0,

where q : ℙ(E) → X is the natural morphism and T_{q} is the relative tangent bundle of q.

104.E. Deformation to the normal cone. Let f : Y → X be a closed embedding of schemes. First suppose that X is an affine scheme, X = Spec(A), and Y is given by an ideal I ⊂ A. Set Y = Spec(A/I). Consider the subring

Ã = ∐_{n∈ℤ} I^{−n}t^{n}

of the Laurent polynomial ring A[t, t^{−1}], where the negative powers of the ideal I are understood as equal to A. The scheme D_{f} := Spec(Ã) is called the deformation scheme of the closed embedding f. In the general case, in order to define D_{f}, we cover X by open affine subschemes and glue together the deformation schemes of the restrictions of f to the open sets of the covering.

The inclusion of A[t] in Ã induces a morphism g : D_{f} → 𝔸^{1} × X. Denote by C_{f} the inverse image g^{−1}({0} × X). In the affine case,

C_{f} = Spec(A/I ⊕ I/I^{2} · t^{−1} ⊕ I^{2}/I^{3} · t^{−2} ⊕ ⋯).

Thus, C_{f} is the normal cone of f (cf. Example 104.3). If f is a regular closed embedding of codimension d, then C_{f} is a vector bundle over Y of rank d. We write N_{f} for C_{f} in this case.

The open complement D_{f} \ C_{f} is the inverse image g^{−1}(𝔾_{m} × X). In the affine case, it is the spectrum of the ring Ã[t^{−1}] = A[t, t^{−1}]. Hence the inverse image is canonically isomorphic to 𝔾_{m} × X via g, i.e.,

D_{f} \ C_{f} ≃ 𝔾_{m} × X.

In the affine case, the natural ring homomorphism A[t] → (A/I)[t] extends canonically to a ring homomorphism Ã → (A/I)[t]. Hence the morphism f × id :

𝔸^{1} × Y → 𝔸^{1} × X factors through the canonical morphism h : 𝔸^{1} × Y → D_{f} over 𝔸^{1}. The fiber of h over a point t ≠ 0 of 𝔸^{1} is naturally isomorphic to the morphism f. The fiber of h over t = 0 is isomorphic to the zero section Y → C_{f} of the normal cone C_{f} of f. Thus we can view h as a family of closed embeddings parameterized by 𝔸^{1} deforming the closed embedding f into the zero section Y → C_{f} as the parameter t “approaches 0”. We have the following diagram of natural morphisms:

Y ⟶ 𝔸^{1} × Y ⟵ 𝔾_{m} × Y
↓  ↓  ↓
C_{f} ⟶ D_{f} ⟵ 𝔾_{m} × X
↓  ↓  ∥
Y ⟶ 𝔸^{1} × X ⟵ 𝔾_{m} × X.

Note that the normal cone C_{f} is the principal divisor in D_{f} of the function t. Consider a fiber product diagram

(104.21)  \begin{matrix}Y′ & ⟶^{f′} & X′ \\ ^{g}↓ & & ↓^{h} \\ Y & ⟶^{f} & X \end{matrix}

where f and f′ are closed embeddings. It induces the fiber product diagram of open and closed embeddings:

(104.22)  \begin{aligned}C_{f′} & ⟶ & D_{f′} & ⟵ & 𝔾_{m} × X′ \\ ^{k}↓ & & ^{l}↓ & & ↓^{id×h} \\ C_{f} & ⟶ & D_{f} & ⟵ & 𝔾_{m} × X. \end{aligned}

Proposition 104.23. In the notation of (104.21), there are natural closed embeddings D_{f′} → D_{f} ×_{X} X′ and C_{f′} → C_{f} ×_{X} X′. These embeddings are isomorphisms if h is flat.

Proof. We may assume that all schemes are affine and h is given by a ring homomorphism A → A′. The scheme Y is defined by an ideal I ⊂ A and Y′ is given by I′ = IA′ ⊂ A′. The natural homomorphism I^{n}⊗_{A}A′ → (I′)^{n} is surjective, hence Ã ⊗_{A} A′ → Ã′ is surjective. Consequently, D_{f′} → D_{f} ×_{X} X′ and C_{f′} → C_{f} ×_{X} X′ are closed embeddings. If A′ is flat over A, the homomorphism I^{n} ⊗_{A} A′ → (I′)^{n} is an isomorphism. □

104.F. Double deformation space. Let A be a commutative ring.

Lemma 104.24. Let I be the ideal of A generated by a regular sequence 𝔞 = (a_{1}, a_{2}, …, a_{d}) and let a ∈ A be such that a + I is not a zero divisor in A/I. If ax ∈ I^{m} for some x ∈ A and m, then x ∈ I^{m}.

Proof. By Proposition 104.9, multiplication by a + I on I^{n}/I^{n+1} is injective for any n. The statement of the lemma follows by induction on m. □

Let 𝔞 = (a_{1}, a_{2}, …, a_{d}) and 𝔟 = (b_{1}, b_{2}, …, b_{e}) be two sequences of elements of A. We write 𝔞 ⊂ 𝔟 if d ≤ e and a_{i} = b_{i} for all i ∈ [1, d]. Clearly, if 𝔞 ⊂ 𝔟 and 𝔟 is regular, so is 𝔞.

Let I ⊂ J be ideals of A. We define the ideals I^{n}J^{m} for n < 0 or m < 0 by

I^{n}J^{m} = \{ \begin{matrix}J^{n+m} & if n < 0, \\ I^{n} & if m < 0. \end{matrix}

Proposition 104.25. Let 𝔞 ⊂ 𝔟 be two regular sequences in a ring A and let I ⊂ J be the ideals of A generated by 𝔞 and 𝔟 respectively. Then

I^{n}J^{m} ∩ I^{n+1} = I^{n+1}J^{m−1},
I^{n}J^{m} ∩ J^{n+m+1} = I^{n}J^{m+1},

for all n and m.

Proof. We prove the first equality. The proof of the second one is similar. We proceed by induction on m. The case m ≤ 1 is clear. Suppose m ≥ 2. As

the inclusion “⊃” is easy, we need to prove that

I^{n}J^{m} ∩ I^{n+1} ⊂ I^{n+1}J^{m−1}.

Let 𝔡 be a sequence such that 𝔞 ⊂ 𝔡 ⊂ 𝔟 and let L be the ideal generated by 𝔡, so I ⊂ L ⊂ J. By descending induction on the length l(𝔡) of the sequence 𝔡, we prove that

(104.26)  I^{n}J^{m} ∩ I^{n+1} ⊂ L^{n+1}J^{m−1}.

When l(𝔡) = l(𝔞), i.e., 𝔡 = 𝔞 and L = I, we get the desired inclusion.

The case l(𝔡) = l(𝔟), i.e., 𝔡 = 𝔟 and L = J is obvious. Let 𝔠 be the sequence satisfying 𝔞 ⊂ 𝔠 ⊂ 𝔡 and l(𝔠) = l(𝔡) − 1. Let K be the ideal generated by 𝔠. We have L = K +aA where a is the last element of the sequence 𝔡. Assuming (104.26), we shall prove that

I^{n}J^{m} ∩ I^{n+1} ⊂ K^{n+1}J^{m−1}.
Let x ∈ I^{n}J^{m} ∩ I^{n+1}. By assumption,

x ∈ L^{n+1}J^{m−1} = ∑_{k=0}^{n+1} a^{n+1−k}K^{k}J^{m−1},

hence

x = ∑_{k=0}^{n+1} a^{n+1−k}x_{k}

for some x_{k} ∈ K^{k}J^{m−1}. For any s ∈ [0, n + 1], set

y_{s} = ∑_{k=0}^{s} a^{s−k}x_{k}.

We claim that y_{s} ∈ K^{s}J^{m−1} for s ∈ [0, n+1]. We prove the claim by induction on s. The case s = 0 is obvious since y_{0} = x_{0} ∈ J^{m−1}. Suppose y_{s} ∈ K^{s}J^{m−1} for some s < n + 1. We have

x = a^{n+1−s}y_{s} + ∑_{k=s+1}^{n+1} a^{n+1−k}x_{k},

where x_{k} ∈ K^{k}J^{m−1} ⊂ K^{s+1} if k ≥ s+1 and x ∈ I^{n+1} ⊂ K^{s+1}. Hence a^{n+1−s}y_{s} ∈ K^{s+1} and therefore y_{s} ∈ K^{s+1} by Lemma 104.24. Thus y_{s} ∈ K^{s}J^{m−1} ∩ K^{s+1}. By the first induction hypothesis, the latter ideal is equal to K^{s+1}J^{m−2} and y_{s} ∈ K^{s+1}J^{m−2}. Since x_{s+1} ∈ K^{s+1}J^{m−1}, we have y_{s+1} = ay_{s} + x_{s+1} ∈ K^{s+1}J^{m−1}. This proves the claim. By the claim, x = y_{n+1} ∈ K^{n+1}J^{m−1}. □

Let Z ⟶^{g} Y ⟶^{f} X be regular closed embeddings. We have closed embeddings

i : (N_{f})|_{Z} → N_{f} and j : N_{g} → N_{fg}.

We shall construct the double deformation scheme D = D_{f,g} and a morphism D → 𝔸^{2} satisfying all of the following:

(1) D|_{𝔸^{1}×𝔾_{m}} = D_{f} × 𝔾_{m}.

(2) D|_{𝔾_{m}×𝔸^{1}} = 𝔾_{m} × D_{fg}.

(3) D|_{𝔸^{1}×{0}} = D_{j}.

(4) D|_{{0}×𝔸^{1}} = D_{i}.

(5) D|_{{0}×{0}} = N_{i} ≃ N_{j}. As in the case of an ordinary deformation space, it suffices to consider the affine

case: X = Spec(A), Y = Spec(A/I), and Z = Spec(A/J), where I ⊂ J are the ideals of A generated by regular sequences. Consider the subring

Â = ∐_{n,m∈ℤ} I^{n}J^{m−n} · t^{−n}s^{−m}

of the Laurent polynomial ring A[t, s, t^{−1}, s^{−1}] and set D = Spec(Â). Since Â contains the polynomial ring A[t, s], there are natural morphisms D → X × 𝔸^{2} → 𝔸^{2}.

We have

Â[s^{−1}] = ∐_{n,m∈ℤ} I^{n} · t^{−n}s^{−m} = ( ∐_{n,m∈ℤ} I^{n} · t^{−n})[s, s^{−1}],
Â[t^{−1}] = ∐_{n,m∈ℤ} J^{m} · t^{−n}s^{−m} = ( ∐_{n,m∈ℤ} J^{m} · s^{−m})[t, t^{−1}].

This proves (1) and (2).

To prove (3) consider the rings

Â/sÂ = ∐_{n,m∈ℤ} [I^{n}J^{m−n}/I^{n}J^{m−n+1}] · t^{−n},
R = ∐_{m∈ℤ} [J^{m}/J^{m+1}] · s^{−m},
S = ∐_{m∈ℤ} [(J^{m} + I)/(J^{m+1} + I)] · s^{−m}.

We have Spec(R) = N_{fg} and Spec(S) = N_{g}. The natural surjection R → S corresponds to the embedding j : N_{g} → N_{fg}.

Let Ĩ = Ker(R → S). By Proposition 104.25, J^{m} ∩ I = IJ^{m−1}, hence

Ĩ = ∐_{m∈ℤ} [IJ^{m−1} + J^{m+1}/J^{m+1}] · s^{−m}

and

Ĩ^{n} = ∐_{m∈ℤ} [I^{n}J^{m−n} + J^{m+1}/J^{m+1}] · s^{−m}.

Therefore, D_{j} is the spectrum of the ring

∐_{m∈ℤ} [I^{n}J^{m−n} + J^{m+1}/J^{m+1}] · t^{−n}s^{−m}.

It follows from Proposition 104.25 that this ring coincides with Â/sÂ, hence (3). To prove (4) consider the ring

Â/tÂ = ∐_{n,m∈ℤ} [I^{n}J^{m−n}/I^{n+1}J^{m−n−1}] · s^{−m}.

The normal bundle N_{f} is the spectrum of the ring

T = ∐_{n∈ℤ} [I^{n}/I^{n+1}] · u^{−m}.

Let J̃ be the ideal of T of the closed subscheme (N_{f})|_{Z}. We have

J̃^{m} = ∐_{n∈ℤ} [I^{n}J^{m} + I^{n+1}/I^{n+1}] · u^{−m}.

The deformation scheme D_{i} is the spectrum of the ring

U = ∐_{n,m∈ℤ} [I^{n}J^{m} + I^{n+1}/I^{n+1}] · u^{−n}s^{−m}.

We define the surjective ring homomorphism φ : Â/tÂ → U taking

(x + I^{n+1}J^{m−n−1}) · t^{−n}s^{−m}  to  (x + I^{n+1}) · u^{−n}s^{−m+n}.

By Proposition 104.25, the map φ is also injective. Hence φ gives the identification (4). Property (5) follows from (3) and (4).

105. Group actions on algebraic schemes

In this section all schemes are quasi-projective over a field F. We write G = {1, σ} for a cyclic group of order 2.

105.A. G-schemes. A G-scheme is a scheme Y together with a G-action on Y. As Y is a quasi-projective scheme, every pair of points of Y belong to an open affine subscheme. It follows that there is an open G-invariant affine covering of such Y. Therefore, in most of the constructions and proofs, we may restrict to the class of affine G-schemes.

In particular, to define a subscheme Y^{G} ⊂ Y for a G-scheme Y, we may assume that Y is affine, i.e., Y = Spec R, where R is a G-algebra, i.e., a commutative F - algebra with G acting on R by F -algebra automorphisms. Consider the ideal I ⊂ R generated by σ(r) − r for all r ∈ R and set

Y^{G} := Spec(R/I).

Example 105.1. For any scheme X over F, the group G acts on X ×X by permutation of the factors. Then (X×X)^{G} coincides with the image of the diagonal closed embedding Δ : X → X × X. Indeed, if X = Spec(A), then X × X = Spec(A ⊗_{F} A) and the ideal I ⊂ A ⊗_{F} A is generated by σ(a ⊗ a′) − a ⊗ a′ = a′ ⊗ a − a ⊗ a′ for all a, a′ ∈ A. Hence it coincides with the kernel of the product map A ⊗ A → A.

Exercise 105.2. Prove that (X × Y)^{G} = X^{G} × Y^{G} for G-schemes X and Y.

Proposition 105.3. Let Y be a G-scheme and let B be the blowup of Y along Y^{G}. Then G acts naturally on B and the subscheme B^{G} of B coincides with the exceptional divisor ℙ(C), where C is the normal cone of the closed embedding Y^{G} → Y.

Proof. We may assume that Y = Spec R for a G-algebra R. Then Y^{G} = Spec(R/I), where I is the ideal of R generated by σ(r) − r for all r ∈ R, and B = Proj S, where S = R ⊕ I ⊕ I^{2} ⊕ · · ·. The scheme B is covered by open G-invariant subschemes Spec(S_{(s)}) with s = σ(r) − r, r ∈ R. The intersection of Spec(S_{(s)}) with the exceptional divisor ℙ(C) is given by the ideal sS_{(s)} in S_{(s)}. Therefore, it suffices to show that the ideal J in S_{(s)} generated by elements of the form σ(x) − x with x ∈ S_{(s)} coincides with sS_{(s)}. Clearly s = σ(r) − r ∈ J. Conversely, let x = \frac{r}{s^{n}} with r ∈ I^{n} be an element in S_{(s)}. As σ acts on I^{n}/I^{n+1} by multiplication by (−1)^{n}, we have t := (−1)^{n}σ(r) − r ∈ I^{n+1} and hence

σ(x) − x = \frac{t}{s^{n}} = s \frac{t}{s^{n+1}} ∈ sS_{(s)}.  □

Let Y be a G-scheme. To define a scheme Y/G we may assume that Y = Spec R. Set

Y/G := Spec(R^{G}).

The natural morphism p : Y → Y/G is called a G-torsor if Y^{G} = ∅.

Example 105.4. Let Y be a G-scheme and let Y′ = Y \ Y^{G}. Then the natural morphism Y′ → Y′/G is a G-torsor.

Proposition 105.5. Let p : Y → Y/G be a G-torsor. Then (1) p_{∗}(O_{Y}) is a locally free O_{Y/G}-module of rank 2.

(2) The morphism p is étale.

Proof. We may assume that Y = Spec(R) for a G-algebra R. As Y^{G} = ∅, the elements s = σ(r) − r with r ∈ R generate the unit ideal of R, i.e., the G-invariant open sets Spec(R_{s}) cover Y. Replacing R by R_{s} we may assume that s is invertible in R. Then {1, r} is a basis of R over R^{G} since for every a ∈ R we have a = f + gr with f = (aσ(r) − σ(a)r)s^{−1} ∈ R^{G} and g = (σ(a) − a)s^{−1} ∈ R^{G}.

Moreover, r is the root of the quadratic polynomial h(t) = t^{2} − bt + c over R^{G} with b = σ(r) + r and c = σ(r)r. As the element h′(r) = 2r − b = −s is invertible in R, the morphism p is étale (cf. [105, Ch. I, Ex. 3.4]). □

Example 105.6. Let p : Y → Y/G be a G-torsor with Y/G = Spec F. Then Y = Spec K, where K is an étale quadratic F -algebra.

Proposition 105.7. Suppose char F ≠ 2. Let Y be a G-scheme and let U = Y \ Y^{G}. Then

(1) The composition Y^{G} → Y ⟶^{p} Y/G is a closed embedding with the complement U/G.

(2) If I ⊂ O_{Y/G} is the sheaf of ideals of Y^{G} in Y/G, then p^{∗}(I) = J^{2}, where J ⊂ O_{Y} is the sheaf of ideals of Y^{G} in Y.

(3) If Y is regular and Y^{G} is a regular divisor in Y, then Y/G is also regular.

Proof. We may assume that Y = Spec R for a G-algebra R.

(1): The composition Y^{G} → Y → Y/G is given by the composition of algebra homomorphisms R^{G} → R → R/I, where I is the ideal of R generated by σ(r) − r for all r ∈ R. The latter composition is surjective as any r ∈ R is the image of (σ(r) + r)/2 ∈ R^{G}.

The scheme U (respectively U/G) is covered by the open sets Spec(R_{s}) (respectively Spec(R_{s^{2}}^{G})) with s = σ(r) − r, r ∈ R. Therefore, the second statement in (1) follows from the equality (R_{s})^{G} = (R^{G})_{s^{2}}.

(2): We need to prove that (I ∩ R^{G})R = I^{2}. The generators s = σ(r) − r of I satisfy σ(s) = −s. Therefore, the ideal I^{2} is generated by G-invariant elements. Hence I^{2} ⊂ (I ∩ R^{G})R. Conversely, let a ∈ I ∩ R^{G}. We can write a = ∑ r_{i}s_{i} with r_{i} ∈ R and s_{i} ∈ I satisfying σ(s_{i}) = −s_{i}. Hence

a = \frac{1}{2}(a + σ(a)) = \frac{1}{2} ∑(r_{i} − σ(r_{i}))s_{i} ∈ I^{2}.

(3): Let z ∈ Y/G and let y ∈ Y be a point above z. Suppose y ∈ Y \ Y^{G}. The natural morphism Y \ Y^{G} → (Y \ Y^{G})/G is a G-torsor by Example 105.4, hence it is flat by Proposition 105.5. As the local ring O_{Y,y} is regular, so is O_{Y/G,z} by [48, Prop. 17.3.3(i)].

Suppose now that y ∈ Y^{G}. We may assume that F is algebraically closed and y is a rational point. The tangent space T_{Y^{G},y} has codimension 1 in T_{Y,y} and coincides with the subspace of all G-invariant vectors in T_{Y,y}. Hence there is a basis ā_{1}, ā_{2}, …, ā_{n} of 𝔪_{Y,y}/𝔪_{Y,y}^{2} satisfying σ(ā_{1}) = −ā_{1} and σ(ā_{i}) = ā_{i} for i ≥ 2. Lift the ā_{i} to a local system of parameters a_{i} ∈ 𝔪_{Y,y} with σ(a_{1}) = −a_{1} and σ(a_{i}) = a_{i} for i ≥ 2. The completion of the local ring O_{Y/G,z} coincides with

Ô_{Y/G,z} = (Ô_{Y,y})^{G} = F [[a_{1}, a_{2}, …, a_{n}]]^{G} = F [[a_{1}^{2}, a_{2}, …, a_{n}]],

and hence is regular. Therefore, z is a regular point on Y/G. □

105.B. The scheme B_{X}. Let X be a scheme. Write B_{X} for the blowup of X^{2} × 𝔸^{1} := X × X × 𝔸^{1} along Δ(X) × {0}. Since the normal cone of Δ(X) × {0} in X^{2} × 𝔸^{1} is T_{X} ⊕ 𝟙 (cf. Proposition 104.7), the projective cone ℙ(T_{X} ⊕ 𝟙) is the exceptional divisor in B_{X} (cf. [45, B.6.6]).

Let G act on X^{2} × 𝔸^{1} by σ(x, x′, t) = (x′, x, −t). If char F ≠ 2, we have

(X^{2} × 𝔸^{1})^{G} = (X^{2})^{G} × (𝔸^{1})^{G} = Δ(X) × {0}.

Set U_{X} = (X^{2} × 𝔸^{1}) \ (Δ(X) × {0}). By Proposition 105.3, the group G acts naturally on B_{X} so that (B_{X})^{G} = ℙ(T_{X} ⊕ 𝟙) and B_{X} \ ℙ(T_{X} ⊕ 𝟙) is canonically isomorphic to U_{X}.

Proposition 105.8. Suppose char F ≠ 2. If X is smooth, then so are B_{X} and B_{X}/G.

Proof. The scheme B_{X} is a blowup of a smooth scheme along smooth center, hence it is smooth. As the scheme ℙ(T_{X} ⊕ 𝟙) = (B_{X})^{G} is a smooth divisor in B_{X}, the second statement follows from Proposition 105.7(3). □

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Notation

(a), 394 (a)_{m}, 394 A ⋆ B, 387 A^{p}(X, K_{q}) (K-cohomology group), 257 A^{op} (opposite algebra), 388 A_{L}, 387 A_{n} (property), 149 A_{p}(X, K_{q}) (K-homology group), 238 B_{X}, 279 C(X) (Rost complex), 218 C(φ) (Clifford algebra), 57 C(𝔛(F), ℤ), 134 C ⊕ C′ (direct sum of cones), 410 C^{+}(φ), 63 C^{∘}, 413 C_{0}(φ) (even Clifford algebra), 57 C_{Q} (conic curve), 194 C_{f} (normal cone), 410 C_{p}(X), 219 C_{p,n}(X), 219 D(t), 116 D(𝔟) (set of nonzero values of 𝔟), 15 D(φ) (set of nonzero values of φ), 52 D[t], 116 D(∞⟨1⟩), 123 D_{f} (deformation scheme), 414 F (T) (rational function field), 71 F (X_{φ}) (function field), 94 F ((t)) (Laurent series field), 123 F (p) (quotient field), 75 F (t) (rational function field), 17 F (x) (residue field of x), 217 F [T] (polynomial ring), 71 F [t] (polynomial ring), 15 F [t_{1}, …, t_{n}] (polynomial ring), 71 F^{×2}, 12 F_{P} (real closure), 128, 384 F_{a}, 53, 388 F_{py} (pythagorean closure), 125 F_{sep} (separable closure), 393 F_{𝔥(φ)−1} (leading field), 104 G(𝔟) (group of similarity factors), 15 G(φ) (group of similarity factors), 52 GC(𝔛(F), ℤ), 184 GP (F) (set of general quadratic Pfister

forms), 53 GP_{n}(F) (set of general quadratic n-fold

Pfister forms), 53 GW_{∗}(F) (graded Witt ring), 28 H(a), 384 H(a_{1}, …, a_{n}), 385 H^{n}(F) (Galois cohomology group), 404 H^{n}(F, M) (Galois cohomology group), 393 H^{n}(Γ, M) (cohomology group), 393 H^{n,i}(F, ℤ/mℤ), 402 H_{n} (property), 173 I(F) (fundamental ideal), 24 I^{n}(F), 24 I^{n}J^{m}, 416 I_{q}(F) (quadratic Witt group), 51 I_{q}(K/F), 99 I_{q}^{n}(F), 53 I_{t}(F), 130 I_{t}^{n}(F), 148 I_{𝔟}(F), 153 I_{red}^{n}(F), 137 J(φ) (J-invariant), 364 K ⇀ L (place), 408 K_{∗}(F) (graded Milnor ring), 398 K_{0}(X), 410 K_{n}(F) (Milnor group), 398 (L/F, b) (quaternion algebra), 389 L(D), 267 L_{c} (canonical line bundle), 412 L_{n}, 198 L_{t} (tautological line bundle), 413 M(−1), 396 M(X) (Chow motive), 296 M(X)(i) (Chow motive), 296 M^{Γ}, 393 N(ρ), 109 N_{f} (normal bundle), 411 P (F) (set of quadratic Pfister forms), 53 P (n), 351 P^{∨} (dual sheaf), 410 P_{n}(F) (set of quadratic n-fold Pfister

forms), 53 R[t, σ], 391 R^{×} (unit group), 12 T (V) (tensor algebra), 57 T_{X} (tangent cone), 410 T_{g} (relative tangent bundle), 411 U_{X}, 278 V (E), 209 V (T), 71 V [T], 71 V [t], 71 V_{Q} (space of pure quaternions), 194 V_{φ} (underlying vector space of φ), 39 W (F) (Witt ring), 20 W (K/F), 99 W^{⊥} (orthogonal complement), 13 W_{t}(F), 126 W_{red}(F) (reduced Witt ring), 137 X_{φ} (projective quadric), 93 X_{φ}(L) (set of L-points), 93 X_{k}, 105 X_{(p)}, 218 Y/G (factor scheme), 419 Y^{G}, 418 Y_{x} (fiber scheme), 222 Z(a), 262

[A] · [B] (product in Br(F) and Ét_{2}(F)), 388

[Z] (cycle of Z), 219 [a, b] (binary quadratic form), 40

[x] (cycle of a point), 261 Br(F) (Brauer group), 388 Br(L/F), 388 Br_{m}(F), 388 CH(Z; Λ), 295 CH^{∗}(X) (graded Chow group), 267 CH^{p}(X) (Chow group of codimension p

classes of cycles), 267 CH_{p}(X) (Chow group of dimension p

classes of cycles on X), 261 char F (characteristic of F), 11 Ét_{2}(F), 388 Λ(V) (exterior algebra of V), 57 Λ^{2}(V) (exterior square of V), 12 Γ_{F} (absolute Galois group), 393 Γ_{f} (graph of a morphism), 292 Gr_{r}(V) (Grassmannian variety), 389 Ind_{K/F} (M), 395 Λ(i) (Tate motive), 296 𝕄_{n}(F) (n × n matrix ring), 12 N_{A} (quadratic norm form), 60, 387 Nrd (reduced norm), 194, 390 ℙ(C) (projective cone), 409 SB(A) (Severi-Brauer variety), 389 𝕋_{n}(F) (n × n upper-triangular matrices),

41 Tr_{A} (trace form), 60, 387 Trd (reduced trace), 194, 390 Z(X; ℙ^{1}), 263, 278 Z_{p}(X) (group of p-dimensional cycles), 261 α : X ⇝ Y (correspondence), 295 α ∪ β (cup-product), 394 α × β (external product), 232 α^{∗}, 294 α_{G}^{2}, 278 α^{t} (transpose of α), 292 α_{∗}, 293 ann_{m}(ρ), 181 ann_{m}(𝔟), 181 ann_{W (F)}(φ), 55 ann_{W (F)}(𝔟), 37 ē_{0}, 25 ē_{1}, 25 ū(F) (ū-invariant), 161 ū_{r}(F) (ū_{r}-invariant), 162 β ∘ α, 292 ⟨D(φ)⟩, 76 O_{X_{φ}}(s), 309 ∩ (intersection of cycles), 317 S^{•} (symmetric algebra), 93, 118 cd_{p}(F) (cohomological p-dimension), 407 χ(Γ_{F}) (character group), 393 χ_{m}(Γ_{F}), 393 clif(φ) (Clifford invariant), 63 deg φ (degree of φ), 104 deg f (degree), 75 deg v (degree), 75 deg (degree homomorphism), 263 deg(A) (degree of A), 388 δ (connecting homomorphism), 240 det 𝔟 (determinant), 12 dim x (dimension of x), 218 disc(φ) (discriminant), 61 div(f) (divisor of f), 219 ind(A) (index of A), 388 κ(x) (residue field of x), 217 ⟨a⟩_{b}, 14 ⟨a⟩_{q}, 40 ⟨a_{1}, …, a_{n}⟩_{b}, 14 ⟨a_{1}, …, a_{n}⟩_{q}, 44 ⟨⟨a⟩⟩_{b}, 24 ⟨⟨a_{1}, …, a_{n}]], 53 ⟨⟨a]], 53 ⟨⟨a_{1}, …, a_{n}⟩⟩_{b}, 24 ⟨⟨a_{1}, …, a_{n}⟩⟩_{q}, 56 \binom{a,b}{F} (quaternion algebra), 390

\left[{a,b \atop F}\right] (quaternion algebra), 390 𝟙 (trivial line bundle), 410 𝔥(n) (height of an integer n), 351 μ_{a}, 197 μ_{m}, 394 ν(F) (torsion-free index), 190 SB_{r}(A) (generalized Severi-Brauer variety),

389 CM_{*}(F, Λ), 299 CM(F), 299 CM(F, Λ), 298 CR(F), 297 CR(F, Λ), 296 CR_{∗}(F), 297 CR_{∗}(F, Λ), 296 Corr_{i}(X, Y; Λ), 295 C(F, Λ), 296 GO(φ) (group of similitudes), 310 PGO(φ) (group of projective similitudes),

310 Sg(C) (total Segre class), 268 Sm (category of smooth complete schemes),

291 Sq^{X} (Steenrod operation of homological

type), 280 Sq_{X} (Steenrod operation of cohomological

type), 285 Sq_{X}^{k}, 285 Sq_{k}^{X}, 281 O(V, φ) (orthogonal group), 355 O^{+}(V, φ) (special orthogonal group), 355 cdim(X) (canonical dimension of X), 367 cdim(C) (canonical dimension of C), 367 cdim_{p}(X) (canonical p-dimension of X),

367 cdim_{p}(C) (canonical p-dimension of C), 367 ord_{R} (order homomorphism), 262 ord_{x}, 262 sg^{C} (Segre homomorphism), 268 \overline{I}^{n}(F), 143 \overline{ann_{m}}(ρ), 181 \overline{ann_{m}}(𝔟), 181 \overline{e}_{n} (nth graded cohomological invariant), 68 ∂ (first residue homomorphism), 81 ∂_{x′}^{x} (residue homomorphism), 218 ∂_{R} (residue homomorphism), 218 ∂_{Z}^{U} (boundary map), 229 ∂_{π} (second residue homomorphism), 81 ∂_{v} (residue homomorphism), 396, 399 ∂_{x} (residue homomorphism), 397, 399 ψ^{⊥} (complementary form), 42 rad φ (quadratic radical of φ), 42 rad 𝔟 (radical of 𝔟), 13 sgn 𝔟 (signature), 123 sgn (signature map), 123 sgn (total signature map), 134 sgn_{P} (signature map), 128 σ_{f} (deformation homomorphism), 235, 240 st(F) (stable range), 181 st_{r}(F) (reduced stability index), 135 supp 𝔟 (support), 134 τ_{v} (hyperplane reflection), 40 \underline{Hom}(M, N), 300 φ ≺≻ ψ, 97 φ ∼ ψ (Witt equivalence), 49 φ ≃ ψ (isometry), 40 φ′ (pure subform), 66 φ|_{W} (restriction), 41 φ_{1} ⊥ φ_{2} (orthogonal sum), 44 φ_{K} (base extension), 42 φ_{k} (kth kernel form), 104 φ_{an} (anisotropic part), 49 φ_{𝔟} (quadratic form associated to 𝔟), 40 φ_{𝔥(φ)−1} (leading form), 104 Î(F), 24 Î^{n}(F), 24 Ŵ(F) (Witt-Grothendieck ring), 19 D̃(𝔟), 15 D̃(φ), 52 D̃(∞⟨1⟩), 123, 383 G̃(φ), 56 ũ(F) (Hasse number), 173 ℘ (Artin-Schreier map), 41, 388 {a_{1}, a_{2}, …, a_{n}} (n-symbol), 28, 219, 398 {a}, 219 aM, 407 b_{n} (Bockstein map), 406 c(E) (total Chern class), 248 c_{i}(E) (Chern class), 248, 270 c_{K/F} (corestriction homomorphism), 395 c_{L/F} (corestriction (norm)

homomorphism), 400, 402 d_{X} (differential of Rost complex), 218 e(E) (Euler class), 243 e(L)^{•} (total Euler class), 268 e_{0}, 25 e_{1}, 25, 62 e_{2}, 29, 63 e_{n} (nth cohomological invariant), 68 e_{y}(g), 223 f^{∗} (leading coefficient), 75 f^{∗} (pull-back homomorphism), 254 f^{⋆} (Gysin homomorphism), 251, 265 f_{∗}, 28 f_{∗} (push-forward homomorphism), 220,

239, 263 g^{∗} (pull-back homomorphism), 223, 239,

263, 407 h (hyperplane section class), 307 h_{F}^{n,m} (norm residue homomorphism), 404 h_{F}^{n} (norm residue homomorphism), 68, 404 i_{0}(X_{φ}) (Witt index), 93 k_{n}(F), 28 l(A) (length of A), 407 l_{A}(M) (length of M), 407 l_{a}, 195 l_{i} (linear subspace class), 307 nφ, 44 n𝔟, 14 r_{K/F} (restriction homomorphism), 21, 51,

395 r_{L/F} (restriction homomorphism), 388,

398, 402 s(E) (total Segre operation), 269 s(F) (level of F), 33 s_{∗} (transfer map), 84 s_{∗}(𝔟) (transfer map), 82 s_{π} (specialization homomorphism), 400 u′(F) (u′-invariant), 166 u(F) (u-invariant), 161 u(L/F) (relative u-invariant), 167 u_{X}, 280 v^{∗} (leading coefficient), 75 v_{2} (2-adic exponent), 335 v_{X}, 279 w (total Stiefel-Whitney map), 29 w_{i} (ith total Stiefel-Whitney map), 30 x <_{P} 0, 383 x >_{P} 0, 383 S^{2}(V) (symmetric square of V), 12 ℍ (hyperbolic plane), 40 ℍ(V) (hyperbolic form on V), 40 ℍ_{λ} (hyperbolic λ-bilinear plane), 12 ℍ_{λ}(V) (hyperbolic λ-bilinear form), 12 ℤ (ring of integers), 21 𝔛(F) (set of orderings), 383 𝔟 ⊗ φ, 51 𝔟′ (pure subform), 34 𝔟|_{W} (restriction), 12 𝔟_{1} ≃ 𝔟_{2} (isometry), 11 𝔟_{1} ⊗ 𝔟_{2} (tensor product), 19 𝔟_{1} ⊥ 𝔟_{2} (orthogonal sum), 14 𝔟_{K} (base extension to K), 13 𝔟_{φ} (polar form), 39 𝔟_{an} (anisotropic part), 19 𝔥(φ) (height of φ), 104 𝔦(𝔟) (Witt index of 𝔟), 19 𝔦_{0}(φ) (Witt index of φ), 49 𝔦_{k}(X), 105 𝔦_{k}(φ), 104 𝔧_{k}(X), 105 𝔧_{k}(φ), 104 CH(X̄^{r}), 313 Gr(φ), 355 \overline{Ch}e(X^{2}), 317 \overline{Ch}e(X̄^{2}), 317 dim_{Izh}, 325 \overline{CH}(X^{r}) (reduced Chow group), 313 \overline{Ch}(G_{∗}), 371 \overline{Ch}(X^{∗}), 316 \overline{Ch}(X^{r}) (modulo 2 reduced Chow group),

313 mult, 327 nil(W (F)) (nilradical of W (F)), 122 zd(W (F)) (zero divisors of W (F)), 122 J_{n}(F), 105

Terminology

absolute Galois group, 393 absolute stability index, 190 additive completion, 296 admissible extension, 125 affine bundle, 242 algebra

étale, 387 quadratic, 387 quaternion, 389 semisimple, 387 separable, 387 central, 388 opposite, 388 split, 387, 388

alternating matrix, 12 anisotropic part

of a bilinear form, 19 of a quadratic form, 49

anisotropic pattern, 349 Arf invariant, 62 Artin-Schreier map, 41, 388 artinian ring, 407 associativity law, 300

base, 300 basic monomial, 360 bilinear form, 11

alternating, 11 anisotropic, 16 anisotropic part of, 19 determinant of, 12 diagonal, 15 dimension of, 11 isotropic, 16 metabolic, 17 nondegenerate, 11, 79 radical of, 13 round, 15 signature of, 123 signed determinant of, 25 skew-symmetric, 11 subform of, 12 support of, 134 symmetric, 11 tensor product of, 19 totally isotropic subspace of, 17 universal, 16 Witt index of, 19

Bilinear Similarity Norm Principle, 87 Bilinear Similarity Theorem, 87 Bilinear Substitution Principle, 72 Bilinear Value Norm Principle, 78 Bilinear Value Theorem, 77 Bilinear Witt Decomposition Theorem, 18 Bockstein map, 406 boolean space, 384 boundary map, 229 Brauer equivalent algebras, 388 Brauer group, 388

canonical p-dimension, 367 dimension, 367 involution, 389 line bundle, 412

Cartan Formula, 288 Cassels-Pfister Theorem, 71 category

of Chow motives, 299

of correspondences, 296

of graded Chow motives, 298

of graded correspondences, 296 cell, 300 chain equivalence of bilinear forms, 21 Chain Lemma, 391 change of field homomorphism, 223 character group, 393 Chern class, 248 Chow group

of codimension p classes of cycles, 267 of dimension p classes of cycles, 261 reduced, 313

Chow motive, 296 Classification Theorem, 150 Clifford algebra, 57

even, 57 Clifford invariant of a quadratic form, 63 clopen set, 384 cocycle relation, 26 cohomological p-dimension, 407 cohomological invariant of a quadratic

Pfister form, 67 cohomology group, 393 cone, 409

normal, 410 projective, 409 tangent, 410

congruent matrices, 12 connecting homomorphism, 240 corestriction homomorphism, 395, 402 correspondence, 291, 295, 326

incidence, 314, 358 multiplicity of, 327 prime, 326

cup-product, 394 cycle

q-primordial, 323 basic, 313 derivative of, 317 diagram of, 318 essence of, 314 essential, 314 integer produced by 1-primordial, 324 minimal, 317, 342 primordial, 320, 342 rational, 313, 362 subcycle of, 314

cycle of a subscheme, 219 cycles

intersection of, 317

deformation homomorphism, 235, 240 deformation scheme, 414 degree

homomorphism, 263 of a cycle, 311 of a subvariety, 311 of an algebra, 388 of a monomial, 75 of a polynomial vector, 75 of a quadratic form, 104

derivative of a cycle, 317 determinant, 12, 63 diagonalizable bilinear form, 15 diagram of a cycle, 318 dimension

of a bilinear form, 11 of a quadratic form, 39

dimension map, 24 discriminant of a quadratic form, 61 divisor, 219 double deformation scheme, 417 duality functor, 299

Euler class, 243 Excess Formula, 251 excess vector bundle, 251 exchange isomorphism, 292 exponent, 388 external product, 232 field

p-generic, 367 p-special, 406 admissible extension of, 125 euclidean, 122, 383 formally real, 74, 383 function field of a quadric, 94 generic, 367 Hasse number of, 173 isotropy, 367 Laurent series, 123 leading field of a quadratic form, 104 linked, 172 of finite stable range, 181 ordered, 383 pythagorean, 123 pythagorean closure of, 125 quadratic closure of, 125 quadratically closed, 121 real closed, 383 reduced stability index of, 135 relative u-invariant of, 167 stable range of, 181

field extension, 383 excellent, 113 splitting, 388

first residue homomorphism, 81 Frobenius Reciprocity, 83 function field of a quadric, 94 fundamental ideal of the Witt ring, 24

G-algebra, 418 Galois module, 393 generic splitting tower, 104 graded Milnor ring, 398 graph of a morphism, 292 group

of p-dimensional cycles, 261 of projective similitudes, 310 of similarity factors, 15, 52 of similitudes, 310

G-scheme, 418 G-torsor, 419 Gysin homomorphism, 251, 265

Harrison subbasis, 384 Hasse number, 173 Hasse Principle

strong, 173 weak, 172

Hasse-Witt invariant, 31 Hauptsatz, 36, 100 height of a quadratic form, 104 height of an integer, 351 Herbrand index, 407 Homotopy Invariance, 242, 265 hyperbolic λ-bilinear form, 12 hyperbolic form

bilinear, 12

quadratic, 40 hyperbolic pair, 12, 40 hyperbolic plane

bilinear, 12

quadratic, 40 hyperplane reflection, 40

idempotent completion of an additive

category, 298 index of an algebra, 388 intersecting properly subvarieties, 266 isometry

of bilinear forms, 11

of quadratic forms, 40 Izhboldin dimension, 325

J-invariant, 364

K-cohomology group, 257 kernel form, 104 K-homology group, 238

lagrangian of a bilinear form, 17 leading coefficient of a polynomial, 75 leading field of a quadratic form, 104 leading form of a quadratic form, 104 leading term of a polynomial, 75 leading vector of a polynomial vector, 75 level of a field, 33 linkage

of bilinear Pfister forms, 37 of quadratic Pfister forms, 102

linkage number of bilinear Pfister forms, 37 Local-Global Principle, 128 localization exact sequence, 240 Localization Sequence, 264

Milnor Conjecture, 68 Milnor’s Theorem, 399 motive, 296, 299 m-periodic module, 396 multiplicity of a correspondence, 327 multiplier, 309

negative with respect to an ordering, 383 Nilpotence Theorem, 302, 373 norm, 387 norm homomorphism, 400 norm residue homomorphism, 404 normal bundle, 411 Normality Theorem, 135

order homomorphism, 262 in a ring, 116

ordering, 383 orientation, 309, 313 orthogonal basis, 14 orthogonal complement, 13 orthogonal subspace, 13 orthogonal sum external, 14, 44 internal, 13, 42 orthogonal vectors, 13

Pfister form n-fold, 24, 53 associated general, 101 bilinear, 24 r-linked, 37 chain p-equivalent, 33 divides, 35 general, 35 linkage number of, 37 linkage of, 37 linked, 37 pure subform of, 34 simply p-equivalent, 33

divisor of, 102 quadratic, 53 r-linked, 102 chain p-equivalent, 65 general, 53 linkage of, 102 linked, 102 neighbor, 101 pure subform of, 66 simply p-equivalent, 65

Pfister neighbor, 101

place, 408 center of, 408 composition, 408 geometric, 408 induced by, 408 surjective, 408 polar form, 39 polar identity, 39 polynomial

σ-twisted, 391 marked, 227

positive with respect to an ordering, 383 preordering, 383 prime cycle, 261 Projection Formula, 258 projective bundle, 410 Projective Bundle Theorem, 246, 265 proper inverse image, 266 Property A_{n}, 149 Property H_{n}, 173 p-special closure, 407 pull-back homomorphism, 223, 239, 254,

263 pure quaternion, 390 pure subform

of a bilinear Pfister form, 34 of a quadratic Pfister form, 66

push-forward homomorphism, 220, 239, 263 pythagoras number, 332 pythagorean closure, 125

Quadratic Cancellation Theorem, 48 quadratic closure, 125 quadratic form, 39

kth kernel form, 104 nth cohomological invariant of, 68 absolute higher Witt index of, 104 anisotopic, 41 anisotropic part of, 49 annihilators of, 55 associated, 40 Clifford invariant of, 63 defined over a field, 111 degree of, 104 determinant of, 63 diagonalizable, 44 dimension of, 39 discriminant of, 61 excellent, 113 height of, 104 irreducible, 94 isotropic, 41 kernel form of, 104 leading field of, 104 leading form of, 104 locally hyperbolic, 161 locally isotropic, 172 nondegenerate, 43 polar form of, 79 regular, 42 regular part of, 42 relative higher Witt index of, 104 round, 52 splits off a hyperbolic plane, 42 splitting pattern of, 104 subform, 41 tensor product of, 51 totally indefinite, 172 totally isotropic subspace of, 41 totally singular, 42 universal, 52 Witt-equivalent, 49

quadratic radical, 42 Quadratic Similarity Norm Principle, 86 Quadratic Similarity Theorem, 86 Quadratic Substitution Principle, 72 Quadratic Value Norm Principle, 78 Quadratic Value Theorem, 76 quadratic Witt group, 51 quadratically closed field, 17 quadric

oriented, 309, 313 projective, 93 split, 307

quasi-Pfister form, 56

radical bilinear, 13

quadratic, 42 ramification index, 223 rationally equivalent cycles, 263 reduced Chow group, 313

modulo 2, 313 reduced norm map, 390 reduced stability index, 135 reduced trace map, 390 regular part of a quadratic form, 42 relative u-invariant, 167 relative dimension, 222 relative tangent bundle, 411 Representation Theorem, 73 represented values, 15, 52 residue homomorphism, 218, 399 restriction homomorphism, 21, 51, 223,

395, 398, 402 Rost complex, 229 Rost correspondence, 341 Rost motive, 379

scheme, 217, 261, 277 rank of, 373 relatively cellular, 300 tractable, 373

second residue homomorphism, 81 Segre closed embedding, 389 Segre homomorphism, 268 Separation Theorem, 108 sequence

regular, 411

regular of codimension d, 411 Severi-Brauer variety, 389

generalized, 389 shell triangle, 320, 321

dual, 321 signature, 123 total, 134

signature map, 123, 128

total, 134 signed determinant, 25, 62 similar

bilinear forms, 12 quadratic forms, 41

similarity factor of a bilinear form, 15 of a quadratic form, 52

similitude, 309

simply chain equivalent forms, 21 space of orderings, 383 specialization, 218 specialization homomorphism, 235, 400 split class, 388 split idempotent, 298 split object, 299 splitting field extension, 388 splitting pattern, 104 Splitting Principle, 247 Springer’s Theorem, 77, 313 stable range, 181 Steenrod operations

of cohomological type, 277, 285 of homological type, 277

Stiefel-Whitney map, 30 Strong Splitting Principle, 275 subform, 12

complementary form of, 42 Subform Theorem, 94 Sylvester’s Law of Inertia, 123 symbol, 398 symmetrization, 344 symplectic basis, 14

tangent bundle, 412 Tate motive, 296, 299 tautological line bundle, 413 tensor algebra, 57 tensor category, 297 tensor product, 19 torsion-free index, 190 total Chern class, 248 total Euler class, 268 total Segre class, 268 total Segre operation, 269 total Stiefel-Whitney class, 29 total Stiefel-Whitney map, 29 totally isotropic subspace, 17 totally negative, 383 totally positive, 383 totally positive elements, 383 trace, 387 tractable class, 373 transfer

of bilinear forms, 82

of quadratic forms, 82 transfer maps

of Witt groups, 84

of Witt rings, 84 transpose of a correspondence, 292

ū-invariant, 161 ū_{r}-invariant, 162 u-invariant, 161 u′-invariant, 166

variety, 217, 261 vector

anisotropic, 16, 41

isotropic, 16, 41

primitive, 79 vector bundle, 410

tautological, 356

weight of a monomial, 361 Weil’s Reciprocity Law, 222 Whitney Sum Formula, 30, 250 Witt Cancellation Theorem, 19 Witt Chain Equivalence Theorem, 22 Witt class, 20 Witt Decomposition Theorem, 48 Witt equivalence, 49 Witt Extension Theorem, 47 Witt group of classes of quadratic forms, 51 Witt index

absolute higher, 104 of a bilinear form, 19 of a quadratic form, 49 of a quadric, 93 relative higher, 104

Witt relations, 22

Witt ring graded, 28 annihilator ideals in, 37 annihilators in, 182 reduced, 137

Witt ring of classes of symmetric bilinear

forms, 20 Witt-Grothendieck ring, 19 Wu Formula, 286

zero section, 410