Quadratic Forms, Linear Algebraic Groups, and Cohomology (eBook)

Preface 6
Part I Surveys 12
Multiples of forms 13
1 Multiples of Quadratic Forms 14
1.1 Galois Cohomology 14
1.2 Quadratic Forms 14
2 Hermitian forms over Division Algebras with Involution 15
3 Galois Cohomology of Unitary Groups 17
4 Systems of Quadratic and Hermitian Forms 19
5 G-Quadratic Forms 19
References 20
On Saltman's p-Adic Curves Papers 22
1 Discrete Valuations 24
2 Residue Map 25
3 Surfaces 27
4 Unramified Brauer Group and Arithmetic Surfaces 29
5 Modified Picard Group 31
6 An Exact Sequence for Local Surfaces 33
7 Computations 34
8 Program for Splitting in Prime Degree 38
9 Splitting in Prime Degree 42
References 48
Serre's Conjecture II: A Survey 49
1 Introduction 49
2 Fields of Cohomological Dimension 2 50
3 Link Between the Conjecture and the Classification of Groups 52
4 Approaches to the Conjecture 53
4.1 Subgroup Trick 54
4.2 Rost Invariant 54
4.3 Serre's Injectivity Question 55
5 Known Cases in Terms of Groups 56
5.1 Classical Groups 56
5.2 Quasi-split Exceptional Groups 56
5.3 Other Exceptional Groups 57
6 Known Cases in Terms of Fields 58
6.1 l-Special Fields 58
6.2 Complete Valued Fields 58
6.3 Global Fields 59
6.4 Function Fields 59
6.5 Why Theorem 6.3 Implies Theorem 6.2 59
7 Remaining Cases and Open Questions 61
References 62
Field Patching, Factorization, and Local--Global Principles 65
1 Introduction 65
2 Patches and Local--Global Principles 66
2.1 Fields Associated to Patches 66
2.2 Some Local--Global Principles 67
3 Patching 69
3.1 Patching Finite Dimensional Vector Spaces 70
3.2 Patching Algebraic Objects 71
3.3 Central Simple Algebras and Quadratic Forms 72
3.4 Properties of R"0362RP,R"0362RU, FP, FU 73
4 Local--Global Principles, Factorization, and Patching 74
4.1 Local--Global Principles for Rational Points 74
4.2 Local--Global Principles for Algebraic Objects and Torsors 76
5 Factorization for Retract Rational Groups 77
5.1 Overview and Preliminaries 77
5.2 Retractions: Basic Definitions and Properties 79
5.3 Adic Convergence of Taylor Series 81
5.4 Factorization 86
5.5 Proof of Lemma 5.2.9 87
References 89
Deformation Theory and Rational Points on Rationally Connected Varieties 91
1 Introduction 91
2 Deformations of Maps and Rational Connectivity 93
3 Deformations of Curves, Stable Maps, the Graber--Harris--Starr Theorem,and Irreducibility of Mg 99
4 Moduli of Porcupines and Rational Sections of Rationally Simply Connected Fibrations over Surfaces 107
References 116
Recent Progress on the Kato Conjecture 117
1 Statements of the Kato Conjectures 117
2 Known Results and Announcement of New Results 119
3 Outline of Proof of Theorem 2.5 122
4 Applications 129
References 131
Elliptic Curves and Iwasawa's µ = 0 Conjecture 133
1 Introduction 133
2 Iwasawa's µ = 
134 
3 Free Pro-p Groups and Iwasawa's µ = 
136 
4 An Analogue for Elliptic Curves 137
References 142
Cohomological Invariants of Central Simple Algebras with Involution 144
1 Introduction: Classification of Quadratic Forms 145
2 From Quadratic Forms to Involutions 147
3 Orthogonal Involutions 149
3.1 The Split Case 150
3.2 The Case of Index 2 151
3.3 Discriminant 153
3.4 Clifford Algebras 154
3.5 Higher Invariants 157
4 Unitary Involutions 160
4.1 The (Quasi)split Case 161
4.2 The Discriminant Algebra 164
4.2.1 The Odd Degree Case 165
4.2.2 The Even Degree Case 165
4.3 Higher Invariants 166
5 Symplectic Involutions 168
5.1 The Case of Index 2 168
5.2 Invariant of Degree 2 170
5.3 The Discriminant 171
Appendix: Trace Form Invariants 175
References 176
Witt Groups of Varieties and the Purity Problem 179
1 The Witt Ring of a Field 179
2 The Witt Ring of a Variety 182
3 Purity 184
4 The Proof of the Purity Theorem 188
References 190
Part II Invited Articles 192
Some Extensions and Applications of the Eisenstein Irreducibility Criterion 193
References 200
On the Kernel of the Rost Invariant for E8 Modulo 3 202
1 Introduction 202
2 Notation and Auxiliary Results 203
2.1 Split Groups 203
2.2 Steinberg's Theorem 204
3 The Rost Invariant and its Properties 205
3.1 Inner Type Ap-1 205
3.2 The Rost Multipliers 206
4 Reduction to Special Cocycles 206
5 3-Sylow Subgroups of the Weyl Group WE 8 208
6 Galois Descent Data for Groups of Type A2 210
7 Construction of 1 211
8 The Direct Product Decomposition of S1 212
9 Construction of 2 214
10 Construction of 3 and proof of (b) 215
References 216
Une version du théorème d'Amer et Brumer pour les zéro-cycles 218
1 Introduction 218
2 Indice et indice réduit 219
3 Système de deux formes 220
4 Système de plusieurs formes, I 222
5 Système de plusieurs formes, II 224
Littérature 225
Quaternion Algebras with the Same Subfields 227
1 Introduction 227
2 An Example 228
3 Discrete Valuations: Good Residue Characteristic 229
4 Discrete Valuations: Bad Residue Characteristic 231
5 Unramified Cohomology 232
6 Transparent Fields 234
7 Proof of the Main Theorem 235
8 Pfister forms and Nondyadic Valuations 236
9 Theorem for Quadratic Forms 237
10 Appendix: Tractable Fields 237
References 239
Lifting of Coefficients for Chow Motives of Quadrics 241
1 Introduction 241
2 Chow Groups of Quadrics 242
3 Lifting of Coefficients 243
4 Surjectivity in the Main Theorem 244
5 Injectivity in the Main Theorem 245
References 248
Upper Motives of Outer Algebraic Groups 250
1 Introduction 250
2 Nilpotence Principle for Quasi-homogeneous Varieties 252
3 Corestriction of Scalars for Motives 253
4 Proof of Theorem 1.1 255
References 258
Triality and étale algebras 259
1 Introduction 259
2 Étale Algebras and -sets 260
3 Cohomology 266
4 Triality and -Coverings 267
5 The Weyl Group of D4 272
6 Triality and étale Algebras 275
7 Trialitarian Resolvents 280
8 Triality and Witt Invariants of étale Algebras 281
References 285
Remarks on Unimodular Rows 287
1 Introduction 287
2 Proof of Theorem 4 288
3 Unimodular Rows of Length more than Three 292
References 293
Vector Bundles Generated by Sections and Morphisms to Grassmannians 294
1 Introduction 294
2 Vector Bundles Generated by Sections 295
3 Morphisms from Projective Space to Gr(2,Ck) 298
References 302
Adams Operations and the Brown-Gersten-Quillen Spectral Sequence 303
1 Introduction 303
2 The Category Al 304
2.1 Definition of Al 304
2.2 A Decomposition 305
3 Spectral Sequences in K-Theory 306
3.1 Adams Operations 306
3.2 Localization Exact Sequence and the Niveau Spectral Sequence 308
3.3 Motivic Spectral Sequence 310
References 310
Non-self-dual Stably Free Modules 312
1 Introduction 312
2 Proof via Clutching Functions 313
3 Proof via Classifying Spaces 314
4 Proof via Homotopy Theory 315
5 Proof via Suslin Matrices 316
6 Proof via Riemann--Roch 318
References 320
Homotopy Invariance of the Sheaf WNis and of Its Cohomology 322
1 Introduction 322
2 Voevodsky Trick 323
3 Auxiliary Results 325
4 Proof of the Main Theorem 326
5 The Case of an Imperfect Ground Field k 330
6 The Case of a Finite Ground Field k 332
References 332
Imbedding Quasi-split Groups in Isotropic Groups 333
1 Introduction 333
2 Classical Groups 334
2.1 Notation 334
2.2 Type 1An 335
2.3 Type Bn, n 2 335
2.4 Groups of Type 2An, Cn, and 1Dn, 2Dn, n > 3
3 Exceptional Groups 338
3.1 Further Notation 338
3.2 340
3.3 Type 1E286,2 340
3.4 341
3.5 341
3.6 Groups of Type 1E166,2 and 2E16''6,2 342
3.7 Groups of Type 2E16'6,2 342
3.8 Groups of Type E317,2 342
3.9 Groups of Type E287,3 and E97,4 343
3.10 Groups of Type E788,2 and E288,4 344
3.11 Groups of Type E668,2 344
References 344