Central Simple Algebras and Galois Cohomology

Gille P., Szamuely T.0511226357

This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. It is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory.

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Table of contents :
Half-title……Page 3
Series-title……Page 6
Title……Page 7
Copyright……Page 8
Contents……Page 9
Preface……Page 13
Acknowledgments……Page 14
1.1 Basic properties……Page 15
1.2 Splitting over a quadratic extension……Page 18
1.3 The associated conic……Page 21
1.4 A theorem of Witt……Page 23
1.5 Tensor products of quaternion algebras……Page 26
2.1 Wedderburn’s theorem……Page 31
2.2 Splitting fields……Page 34
2.3 Galois descent……Page 38
2.4 The Brauer group……Page 43
2.5 Cyclic algebras……Page 47
2.6 Reduced norms and traces……Page 51
2.7 A basic exact sequence……Page 54
2.8 K1 of central simple algebras……Page 56
3.1 Definition of cohomology groups……Page 64
3.2 Explicit resolutions……Page 70
3.3 Relation to subgroups……Page 74
3.4 Cup-products……Page 82
4.1 Profinite groups and Galois groups……Page 94
4.2 Cohomology of profinite groups……Page 99
4.3 The cohomology exact sequence……Page 104
4.4 The Brauer group revisited……Page 109
4.5 Index and period……Page 114
4.6 The Galois symbol……Page 120
4.7 Cyclic algebras and symbols……Page 123
5 Severi–Brauer varieties……Page 128
5.1 Basic properties……Page 129
5.2 Classification by Galois cohomology……Page 131
5.3 Geometric Brauer equivalence……Page 134
5.4 Amitsur’s theorem……Page 139
5.5 An application: making central simple algebras cyclic……Page 145
6.1 Cohomological dimension……Page 149
6.2 C1-fields……Page 154
6.3 Cohomology of Laurent series fields……Page 160
6.4 Cohomology of function fields of curves……Page 165
6.5 Application to class field theory……Page 171
6.6 Application to the rationality problem: the method……Page 174
6.7 Application to the rationality problem: the example……Page 181
6.8 Residue maps with finite coefficients……Page 185
6.9 The Faddeev sequence with finite coefficients……Page 190
7.1 The tame symbol……Page 197
7.2 Milnor’s exact sequence and the Bass-Tate lemma……Page 203
7.3 The norm map……Page 209
7.4 Reciprocity laws……Page 218
7.5 Applications to the Galois symbol……Page 224
7.6 The Galois symbol over number fields……Page 229
8.1 Gersten complexes in Milnor K-theory……Page 237
8.2 Properties of Gersten complexes……Page 241
8.3 A property of Severi-Brauer varieties……Page 246
8.4 Hilbert’s Theorem 90 for K2……Page 252
8.5 The Merkurjev-Suslin theorem: a special case……Page 259
8.6 The Merkurjev-Suslin theorem: the general case……Page 264
9.1 The theorems of Teichmüller and Albert……Page 273
9.2 Differential forms and p-torsion in the Brauer group……Page 280
9.3 Logarithmic differentials and flat p-connections……Page 283
9.4 Decomposition of the de Rham complex……Page 290
9.5 The Bloch–Gabber–Kato theorem: statement and reductions……Page 293
9.6 Surjectivity of the differential symbol……Page 296
9.7 Injectivity of the differential symbol……Page 302
A.1 Affine and projective varieties……Page 312
A.2 Maps between varieties……Page 314
A.3 Function fields and dimension……Page 316
A.4 Divisors……Page 319
A.5 Complete local rings……Page 322
A.6 Discrete valuations……Page 324
A.7 Derivations……Page 328
A.8 Differential forms……Page 332
Bibliography……Page 337
Index……Page 353