Introduction to Galois cohomology and its applications

Cover......Page 1
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 13
Introduction......Page 15
Part I An introduction to Galois cohomology......Page 25
I.1 Reminiscences on field theory......Page 27
I.2.1 Definitions and first examples......Page 31
I.2.2 The Galois correspondence......Page 32
I.2.3 Morphisms of Galois extensions......Page 33
I.2.4 The Galois group as a profinite group......Page 35
Exercises......Page 38
II.3.1 Definitions......Page 40
II.3.2 Functoriality......Page 50
II.3.3 Cohomology sets as a direct limit......Page 55
II.4 Cohomology sequences......Page 59
II.4.1 The case of a subgroup......Page 60
II.4.2 The case of a normal subgroup......Page 65
II.4.3 The case of a central subgroup......Page 66
II.5 Twisting......Page 70
II.6 Cup-products......Page 75
Exercises......Page 78
III.7.1 Digression: categories and functors......Page 83
III.7.2 Algebraic group-schemes......Page 91
III.7.3 The Galois cohomology functor......Page 99
III.8 Abstract Galois descent......Page 110
III.8.1 Matrices reloaded......Page 111
III.8.2 Actions of group-valued functors......Page 113
III.8.3 Twisted forms......Page 115
III.8.4 The Galois descent condition......Page 117
III.8.5 Stabilizers......Page 118
III.8.6 Galois descent lemma......Page 120
III.8.7 Hilbert\'s Theorem 90......Page 124
III.9.1 Galois descent of algebras......Page 131
III.9.2 The conjugacy problem......Page 135
III.9.3 Cup-products with values in μ2......Page 140
Exercises......Page 144
IV.10.1 Quadratic forms over rings......Page 148
IV.10.2 Orthogonal groups......Page 150
IV.10.3 Clifford groups and spinors......Page 153
IV.11.1 Galois cohomology of orthogonal groups......Page 159
IV.11.2 Galois cohomology of spinors......Page 161
IV.12.1 Classification of quadratic forms over Q......Page 166
IV.12.2 Higher cohomological invariants......Page 168
Exercises......Page 172
V.13 Étale algebras......Page 174
V.14.1 Definition and first properties......Page 178
V.14.2 Galois algebras and Galois cohomology......Page 184
Exercises......Page 191
VI.15 Group extensions......Page 193
VI.16 Galois embedding problems......Page 199
Exercises......Page 201
Part II Applications......Page 203
VII Galois embedding problems and the trace form......Page 205
VII.17 The trace form of an étale algebra......Page 206
VII.18 Computation of e*(sn)......Page 210
VII.19 Applications to inverse Galois theory......Page 214
Exercises......Page 219
VIII.20 Central simple algebras......Page 221
VIII.21.1 Basic concepts......Page 231
VIII.21.2 Hyperbolic involutions......Page 236
VIII.21.3 Similitudes......Page 240
VIII.21.4 Cohomology of algebras with involution......Page 242
VIII.21.5 Trace forms......Page 245
Exercises......Page 259
IX.22 Reminiscences on schemes......Page 263
IX.23 Torsors......Page 267
Exercises......Page 273
X.24 Formulation of Noether\'s problem......Page 275
X.25 The strategy......Page 276
X.26 Residue maps......Page 279
X.27 An unramified cohomological invariant......Page 285
X.28 Proof of Theorem X.24.1......Page 286
XI The rationality problem for adjoint algebraic groups......Page 288
XI.29 R-equivalence groups......Page 289
XI.30 The rationality problem for adjoint groups......Page 292
XI.31 Examples of non-rational adjoint groups......Page 295
Exercises......Page 301
XII.32 Essential dimension: definition and first examples......Page 304
XII.33 First results......Page 306
XII.34 Cohomological invariants and essential dimension......Page 310
XII.35 Generic objects and essential dimension......Page 312
XII.36 Generically free representations......Page 314
XII.37 Some examples......Page 315
XII.38 Complements and open problems......Page 320
Exercises......Page 322
References......Page 324
Index......Page 328