Garibaldi / Colliot-Thélène / Sujatha Quadratic Forms, Linear Algebraic Groups, and Cohomology

1. Auflage 2010
ISBN: 978-1-4419-6211-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 18, 348 Seiten

Reihe: Developments in Mathematics

The invited papers collected in this volume address topics related to the research of Raman Parimala (plenary speaker at the upcoming ICM 2010). These themes focus primarily on the interplay between algebra, number theory, and algebraic geometry. The included contributions cover exciting research in areas such as field patching and a proof of the Serre`s Conjecture II for function fields of complex surfaces. TOC:- Foreword.- Surveys.- Multiples of forms (Eva Bayer-Fluckiger).- On Saltman`s p-adic curves papers (Eric Brussel).- Serre`s Conjecture II: a survey (Philippe Gille).- Field patching, factorization, and local-global principles (Daniel Krashen).- Deformation theory and rational points on rationally connected varieties (Max Lieblich).- Recent progress on the Kato Conjecture (Shuji Saito).- Elliptic curves and Iwasawa`s µ=0 conjecture (R. Sujatha).- Cohomological invariants of central simple algebras with involution (Jean-Pierre Tignol).- Witt groups of varieties and the purity problem (Kirill Zainoulline).- Invited articles.- Some extensions and applications of Eisenstein irreducibility criterion (Anuj Bishnoi and Sudesh K. Khanduja).- On the kernel of the Rost invariant for E8 modulo 3 (V. Chernousov).- Une version du théorème d`Amer et Brumer pour les zéro-cycles (Jean-Louis Colliot-Thélène and Marc Levine).- Quaternion algebras with the same subfields (Skip Garibaldi and David J. Saltman).- Lifting of coefficients for Chow motives of quadrics (Olivier Haution).- Upper motives of outer algebraic groups (Nikita Karpenko).- Triality and étale algebras (Max-Albert Knus and Jean-Pierre Tignol).- Vector bundles generated by sections and morphisms to Grassmannians (F. Laytimi and D.S. Nagaraj).- Adams operations and the Brown-Gersten-Quillen spectral sequence (Alexander Merkurjev).- Remarks on unimodular rows (N. Mohan Kumar and M. Pavaman Murthy).- Non-self-dual stably free modules (Madhav V. Nori, Ravi A. Rao, and Richard G. Swan).- Homotopy invariance of the sheaf WNis and of its cohomology (I. Panin).- Imbedding quasi-split groups in isotropic groups (M.S. Raghunathan).

Autoren/Hrsg.

Colliot-Thélène, Jean-Louis

Garibaldi, Skip

Sujatha, R.

Suresh, Venapally

Inhaltsverzeichnis

1;Preface;6
2;Part I Surveys;12
2.1;Multiples of forms;13
2.1.1;1 Multiples of Quadratic Forms;14
2.1.1.1;1.1 Galois Cohomology;14
2.1.1.2;1.2 Quadratic Forms;14
2.1.2;2 Hermitian forms over Division Algebras with Involution;15
2.1.3;3 Galois Cohomology of Unitary Groups;17
2.1.4;4 Systems of Quadratic and Hermitian Forms;19
2.1.5;5 G-Quadratic Forms;19
2.1.6;References;20
2.2;On Saltman's p-Adic Curves Papers;22
2.2.1;1 Discrete Valuations;24
2.2.2;2 Residue Map;25
2.2.3;3 Surfaces;27
2.2.4;4 Unramified Brauer Group and Arithmetic Surfaces;29
2.2.5;5 Modified Picard Group;31
2.2.6;6 An Exact Sequence for Local Surfaces;33
2.2.7;7 Computations;34
2.2.8;8 Program for Splitting in Prime Degree;38
2.2.9;9 Splitting in Prime Degree;42
2.2.10;References;48
2.3;Serre's Conjecture II: A Survey;49
2.3.1;1 Introduction;49
2.3.2;2 Fields of Cohomological Dimension 2;50
2.3.3;3 Link Between the Conjecture and the Classification of Groups;52
2.3.4;4 Approaches to the Conjecture;53
2.3.4.1;4.1 Subgroup Trick;54
2.3.4.2;4.2 Rost Invariant;54
2.3.4.3;4.3 Serre's Injectivity Question;55
2.3.5;5 Known Cases in Terms of Groups;56
2.3.5.1;5.1 Classical Groups;56
2.3.5.2;5.2 Quasi-split Exceptional Groups;56
2.3.5.3;5.3 Other Exceptional Groups;57
2.3.6;6 Known Cases in Terms of Fields;58
2.3.6.1;6.1 l-Special Fields;58
2.3.6.2;6.2 Complete Valued Fields;58
2.3.6.3;6.3 Global Fields;59
2.3.6.4;6.4 Function Fields;59
2.3.6.5;6.5 Why Theorem 6.3 Implies Theorem 6.2;59
2.3.7;7 Remaining Cases and Open Questions;61
2.3.8;References;62
2.4;Field Patching, Factorization, and Local--Global Principles;65
2.4.1;1 Introduction;65
2.4.2;2 Patches and Local--Global Principles;66
2.4.2.1;2.1 Fields Associated to Patches;66
2.4.2.2;2.2 Some Local--Global Principles;67
2.4.3;3 Patching;69
2.4.3.1;3.1 Patching Finite Dimensional Vector Spaces;70
2.4.3.2;3.2 Patching Algebraic Objects;71
2.4.3.3;3.3 Central Simple Algebras and Quadratic Forms;72
2.4.3.4;3.4 Properties of R"0362RP,R"0362RU, FP, FU;73
2.4.4;4 Local--Global Principles, Factorization, and Patching;74
2.4.4.1;4.1 Local--Global Principles for Rational Points;74
2.4.4.2;4.2 Local--Global Principles for Algebraic Objects and Torsors;76
2.4.5;5 Factorization for Retract Rational Groups;77
2.4.5.1;5.1 Overview and Preliminaries;77
2.4.5.2;5.2 Retractions: Basic Definitions and Properties;79
2.4.5.3;5.3 Adic Convergence of Taylor Series;81
2.4.5.4;5.4 Factorization;86
2.4.5.5;5.5 Proof of Lemma 5.2.9;87
2.4.6;References;89
2.5;Deformation Theory and Rational Points on Rationally Connected Varieties;91
2.5.1;1 Introduction;91
2.5.2;2 Deformations of Maps and Rational Connectivity;93
2.5.3;3 Deformations of Curves, Stable Maps, the Graber--Harris--Starr Theorem,and Irreducibility of Mg;99
2.5.4;4 Moduli of Porcupines and Rational Sections of Rationally Simply Connected Fibrations over Surfaces;107
2.5.5;References;116
2.6;Recent Progress on the Kato Conjecture;117
2.6.1;1 Statements of the Kato Conjectures;117
2.6.2;2 Known Results and Announcement of New Results;119
2.6.3;3 Outline of Proof of Theorem 2.5;122
2.6.4;4 Applications;129
2.6.5;References;131
2.7;Elliptic Curves and Iwasawa's µ = 0 Conjecture;133
2.7.1;1 Introduction;133
2.7.2;2 Iwasawa's µ =
0 Conjecture;134
2.7.3;3 Free Pro-p Groups and Iwasawa's µ =
0 Conjecture;136
2.7.4;4 An Analogue for Elliptic Curves;137
2.7.5;References;142
2.8;Cohomological Invariants of Central Simple Algebras with Involution;144
2.8.1;1 Introduction: Classification of Quadratic Forms;145
2.8.2;2 From Quadratic Forms to Involutions;147
2.8.3;3 Orthogonal Involutions;149
2.8.3.1;3.1 The Split Case;150
2.8.3.2;3.2 The Case of Index 2;151
2.8.3.3;3.3 Discriminant;153
2.8.3.4;3.4 Clifford Algebras;154
2.8.3.5;3.5 Higher Invariants;157
2.8.4;4 Unitary Involutions;160
2.8.4.1;4.1 The (Quasi)split Case;161
2.8.4.2;4.2 The Discriminant Algebra;164
2.8.4.2.1;4.2.1 The Odd Degree Case;165
2.8.4.2.2;4.2.2 The Even Degree Case;165
2.8.4.3;4.3 Higher Invariants;166
2.8.5;5 Symplectic Involutions;168
2.8.5.1;5.1 The Case of Index 2;168
2.8.5.2;5.2 Invariant of Degree 2;170
2.8.5.3;5.3 The Discriminant;171
2.8.6;Appendix: Trace Form Invariants;175
2.8.7;References;176
2.9;Witt Groups of Varieties and the Purity Problem;179
2.9.1;1 The Witt Ring of a Field;179
2.9.2;2 The Witt Ring of a Variety;182
2.9.3;3 Purity;184
2.9.4;4 The Proof of the Purity Theorem;188
2.9.5;References;190
3;Part II Invited Articles;192
3.1;Some Extensions and Applications of the Eisenstein Irreducibility Criterion;193
3.1.1;References;200
3.2;On the Kernel of the Rost Invariant for E8 Modulo 3;202
3.2.1;1 Introduction;202
3.2.2;2 Notation and Auxiliary Results;203
3.2.2.1;2.1 Split Groups;203
3.2.2.2;2.2 Steinberg's Theorem;204
3.2.3;3 The Rost Invariant and its Properties;205
3.2.3.1;3.1 Inner Type Ap-1;205
3.2.3.2;3.2 The Rost Multipliers;206
3.2.4;4 Reduction to Special Cocycles;206
3.2.5;5 3-Sylow Subgroups of the Weyl Group WE 8;208
3.2.6;6 Galois Descent Data for Groups of Type A2;210
3.2.7;7 Construction of 1;211
3.2.8;8 The Direct Product Decomposition of S1;212
3.2.9;9 Construction of 2;214
3.2.10;10 Construction of 3 and proof of (b);215
3.2.11;References;216
3.3;Une version du théorème d'Amer et Brumer pour les zéro-cycles;218
3.3.1;1 Introduction;218
3.3.2;2 Indice et indice réduit;219
3.3.3;3 Système de deux formes;220
3.3.4;4 Système de plusieurs formes, I;222
3.3.5;5 Système de plusieurs formes, II;224
3.3.6;Littérature;225
3.4;Quaternion Algebras with the Same Subfields;227
3.4.1;1 Introduction;227
3.4.2;2 An Example;228
3.4.3;3 Discrete Valuations: Good Residue Characteristic;229
3.4.4;4 Discrete Valuations: Bad Residue Characteristic;231
3.4.5;5 Unramified Cohomology;232
3.4.6;6 Transparent Fields;234
3.4.7;7 Proof of the Main Theorem;235
3.4.8;8 Pfister forms and Nondyadic Valuations;236
3.4.9;9 Theorem for Quadratic Forms;237
3.4.10;10 Appendix: Tractable Fields;237
3.4.11;References;239
3.5;Lifting of Coefficients for Chow Motives of Quadrics;241
3.5.1;1 Introduction;241
3.5.2;2 Chow Groups of Quadrics;242
3.5.3;3 Lifting of Coefficients;243
3.5.4;4 Surjectivity in the Main Theorem ;244
3.5.5;5 Injectivity in the Main Theorem;245
3.5.6;References;248
3.6;Upper Motives of Outer Algebraic Groups;250
3.6.1;1 Introduction;250
3.6.2;2 Nilpotence Principle for Quasi-homogeneous Varieties;252
3.6.3;3 Corestriction of Scalars for Motives;253
3.6.4;4 Proof of Theorem 1.1;255
3.6.5;References;258
3.7;Triality and étale algebras;259
3.7.1;1 Introduction;259
3.7.2;2 Étale Algebras and -sets;260
3.7.3;3 Cohomology;266
3.7.4;4 Triality and -Coverings;267
3.7.5;5 The Weyl Group of D4;272
3.7.6;6 Triality and étale Algebras;275
3.7.7;7 Trialitarian Resolvents;280
3.7.8;8 Triality and Witt Invariants of étale Algebras;281
3.7.9;References;285
3.8;Remarks on Unimodular Rows;287
3.8.1;1 Introduction;287
3.8.2;2 Proof of Theorem 4;288
3.8.3;3 Unimodular Rows of Length more than Three;292
3.8.4;References;293
3.9;Vector Bundles Generated by Sections and Morphisms to Grassmannians;294
3.9.1;1 Introduction;294
3.9.2;2 Vector Bundles Generated by Sections;295
3.9.3;3 Morphisms from Projective Space to Gr(2,Ck);298
3.9.4;References;302
3.10;Adams Operations and the Brown-Gersten-Quillen Spectral Sequence;303
3.10.1;1 Introduction;303
3.10.2;2 The Category Al;304
3.10.2.1;2.1 Definition of Al;304
3.10.2.2;2.2 A Decomposition;305
3.10.3;3 Spectral Sequences in K-Theory;306
3.10.3.1;3.1 Adams Operations;306
3.10.3.2;3.2 Localization Exact Sequence and the Niveau Spectral Sequence;308
3.10.3.3;3.3 Motivic Spectral Sequence;310
3.10.4;References;310
3.11;Non-self-dual Stably Free Modules;312
3.11.1;1 Introduction;312
3.11.2;2 Proof via Clutching Functions;313
3.11.3;3 Proof via Classifying Spaces;314
3.11.4;4 Proof via Homotopy Theory;315
3.11.5;5 Proof via Suslin Matrices;316
3.11.6;6 Proof via Riemann--Roch;318
3.11.7;References;320
3.12;Homotopy Invariance of the Sheaf WNis and of Its Cohomology;322
3.12.1;1 Introduction;322
3.12.2;2 Voevodsky Trick;323
3.12.3;3 Auxiliary Results;325
3.12.4;4 Proof of the Main Theorem;326
3.12.5;5 The Case of an Imperfect Ground Field k;330
3.12.6;6 The Case of a Finite Ground Field k;332
3.12.7;References;332
3.13;Imbedding Quasi-split Groups in Isotropic Groups;333
3.13.1;1 Introduction;333
3.13.2;2 Classical Groups;334
3.13.2.1;2.1 Notation;334
3.13.2.2;2.2 Type 1An;335
3.13.2.3;2.3 Type Bn, n 2;335
3.13.2.4;2.4 Groups of Type 2An, Cn, and 1Dn, 2Dn, n > 3;336
3.13.3;3 Exceptional Groups;338
3.13.3.1;3.1 Further Notation;338
3.13.3.2;3.2 ;340
3.13.3.3;3.3 Type 1E286,2;340
3.13.3.4;3.4 ;341
3.13.3.5;3.5 ;341
3.13.3.6;3.6 Groups of Type 1E166,2 and 2E16''6,2;342
3.13.3.7;3.7 Groups of Type 2E16'6,2;342
3.13.3.8;3.8 Groups of Type E317,2;342
3.13.3.9;3.9 Groups of Type E287,3 and E97,4;343
3.13.3.10;3.10 Groups of Type E788,2 and E288,4;344
3.13.3.11;3.11 Groups of Type E668,2;344
3.13.4;References;344