The algebraic and geometric theory of quadratic forms

444 Pages·2008·1.895 MB·English

by  Elman R., Karpenko N., Merkurjev A.

The Algebraic and Geometric Theory
of Quadratic Forms
Richard Elman
Nikita Karpenko
Alexander Merkurjev
Department of Mathematics, University of California, Los Ange-
les, CA 90095-1555, USA
E-mail address: [email protected]
Institut de Mathe´matiques de Jussieu, Universite´ Pierre et Marie
Curie - Paris 6, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE
E-mail address: [email protected]
Department of Mathematics, University of California, Los Ange-
les, CA 90095-1555, USA
E-mail address: [email protected]

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
(cid:161) (cid:162)
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 (F) 103
n
26. The Separation Theorem 107
27. A further characterization of quadratic Pfister forms 109
28. Excellent quadratic forms 111
v
vi CONTENTS
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 In(F) and torsion Pfister forms 147
Chapter VI. u-invariants 161
36. The u¯-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 In(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 208
2
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
CONTENTS vii
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 X2 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 In 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
(cid:161) (cid:162)
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 -fields 385
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viii CONTENTS
98. Algebras 387
99. Galois cohomology 393
100. Milnor K-theory of fields 397
101. The cohomology groups Hn,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
theoryofsymmetricbilinearformsandthetheoryofquadraticformsareessentially
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. Fromtheviewpointofringtheory,Wittgaveapresentationofthisringasa
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 theo-
retic 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
sumofsquaresandnonformallyrealfields. Inparticular,theKrulldimensionofthe
Wittringisoneintheformallyrealcaseandzerootherwise. Thismakesthestudy
of the interaction of bilinear forms and orderings an imperative, hence the impor-
tanceoflookingatrealclosuresofthebasefieldresultinginextensionsofSylvester’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
2n squares. Pfister also used it to show that in a nonformally real field, the least
numbers(F)sothat−1isasumofs(F)squaresisalwaysapowerof2(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
idealI(F)intheWittringofF,calledthefundamentalideal. ItspowersIn(F):=
(cid:161) (cid:162)
n
I(F) , each generated by appropriate Pfister forms, give an important filtration
of W(F). The problem then arises: What ring theoretic properties respect this
1
2 INTRODUCTION
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 (F) of a field (cf. [98]), Milnor gave an
2 (cid:76)
ad hoc definition of a graded ring K (F):= K (F) of a field in [106]. From
∗ n≥0 n
the viewpoint of Galois cohomology, this was of great interest as there is a natural
map, called the norm residue map, from K (F) to the Galois cohomology group
n
Hn(Γ ,µ⊗n)whereΓ istheabsoluteGaloisgroupofF and misrelativelyprime
F m F
to the characteristic of F. For the case m=2, Milnor conjectured this map to be
anepimorphismwithkernel2K (F)foralln. Voevodskyprovedthisconjecturein
n
[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.
Inthisbook,wedonotrestrictourselvestofieldsofcharacteristicdifferentfrom
2. Historically the cases of fields of characteristic different from 2 and 2 have been
studiedseparately. Usuallythecaseofcharacteristicdifferentfrom2isinvestigated
first. In this book, we shall give characteristic free proofs whenever possible. This
meansthatthestudyofsymmetricbilinearformsandthestudyofquadraticforms
mustbedoneseparately,theninterrelated. Wenotonlypresenttheclassicaltheory
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
beendone. Indeed,thefinaltwopartsofthebookwillbebasedonsuchaviewpoint.
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.
Notonlyaretheassociatedvarietiesimportantbutalsothemorphismsbetween
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 fur-
ther 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 homomor-
phisms, 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(cid:48) and B(cid:48) for
the existence of certain cycles on certain varieties. Starting with cycles satisfying