Galois Cohomology

By: Jean-Pierre Serre, J. P. Serre

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Hardcover | 1 January 1997 | Edition Number 2

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This volume is an English translation of "Cohomologie Galoisienne" . The original edition (Springer LN5, 1964) was based on the notes, written with the help of Michel Raynaud, of a course I gave at the College de France in 1962-1963. In the present edition there are numerous additions and one suppression: Verdier's text on the duality of profinite groups. The most important addition is the photographic reproduction of R. Steinberg's "Regular elements of semisimple algebraic groups", Publ. Math. LH.E.S., 1965. I am very grateful to him, and to LH.E.S., for having authorized this reproduction. Other additions include: - A proof of the Golod-Shafarevich inequality (Chap. I, App. 2). - The "resume de cours" of my 1991-1992 lectures at the College de France on Galois cohomology of k(T) (Chap. II, App.). - The "resume de cours" of my 1990-1991 lectures at the College de France on Galois cohomology of semisimple groups, and its relation with abelian cohomology, especially in dimension 3 (Chap. III, App. 2). The bibliography has been extended, open questions have been updated (as far as possible) and several exercises have been added. In order to facilitate references, the numbering of propositions, lemmas and theorems has been kept as in the original 1964 text. Jean-Pierre Serre Harvard, Fall 1996 Table of Contents Foreword ........................................................ V Chapter I. Cohomology of profinite groups 1. Profinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . .

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Non-Fiction Mathematics Number Theory Groups & Group Theory Algebra Geometry Algebraic Geometry

Foreword	p. V
Cohomology of profinite groups
Profinite groups	p. 3
Definition	p. 3
Subgroups	p. 4
Indices	p. 5
Pro-p-groups and Sylow p-subgroups	p. 6
Pro-p-groups	p. 7
Cohomology	p. 10
Discrete G-modules	p. 10
Cochains, cocycles, cohomology	p. 10
Low dimensions	p. 11
Functoriality	p. 12
Induced modules	p. 13
Complements	p. 14
Cohomological dimension	p. 17
p-cohomological dimension	p. 17
Strict cohomological dimension	p. 18
Cohomological dimension of subgroups and extensions	p. 19
Characterization of the profinite groups G such that cd_p(G) ≤ 1	p. 121
Dualizing modules	p. 24
Cohomology of pro-p-groups	p. 27
Simple modules	p. 27
Interpretation of H¹: generators	p. 29
Interpretation of H²: relations	p. 33
A theorem of Shafarevich	p. 34
Poincaré groups	p. 38
Nonabelian cohomology	p. 45
Definition of H⁰ and of H¹	p. 45
Principal homogeneous spaces over A - a new definition of H¹(G, A)	p. 46
Twisting	p. 47
The cohomology exact sequence associated to a subgroup	p. 50
Cohomology exact sequence associated to a normal subgroup	p. 51
The case of an abelian normal subgroup	p. 53
The case of a central subgroup	p. 54
Complements	p. 56
A property of groups with cohomological dimension &leq 1	p. 57
Bibliographic remarks for Chapter I	p. 60
J. Tate - Some duality theorems	p. 61
The Golod-Shafarevich inequality	p. 66
The statement	p. 66
Proof	p. 67
Galois cohomology, the commutative case
Generalities	p. 71
Galois cohomology	p. 71
First examples	p. 72
Criteria for cohomological dimension	p. 74
An auxiliary result	p. 74
Case when p is equal to the characteristic	p. 75
Case when p differs from the characteristic	p. 76
Fields of dimension ≤ 1	p. 78
Definition	p. 78
Relation with the property (C₁)	p. 79
Examples of fields of dimension &leq 1	p. 80
Transition theorems	p. 83
Algebraic extensions	p. 83
Transcendental extensions	p. 83
Local fields	p. 85
Cohomological dimension of the Galois group of an algebraic number field	p. 87
Property (C_r)	p. 87
p-adic fields	p. 90
Summary of known results	p. 90
Cohomology of finite G_k-modules	p. 90
First applications	p. 93
The Euler-Poincaré characteristic (elementary case)	p. 93
Unramified cohomology	p. 94
The Galois group of the maximal p-extension of k	p. 95
Euler-Poincaré characteristics	p. 99
Groups of multiplicative type	p. 102
Algebraic number fields	p. 105
Finite modules - definition of the groups Pⁱ(k, A)	p. 105
The finiteness theorem	p. 106
Statements of the theorems of Poitou and Tate	p. 107
Bibliographic remarks for Chapter II	p. 109
Galois cohomology of purely transcendental extensions	p. 110
An exact sequence	p. 110
The local case	p. 1ll
Algebraic curves and function fields in one variable	p. 112
The case K= k(T)	p. 113
Notation	p. 114
Killing by base change	p. 115
Manin conditions, weak approximation and Schinzel's hypothesis	p. 116
Sieve bounds	p. 117
Nonabelian Galois cohomology
Forms	p. 121
Tensors	p. 121
Examples	p. 123
Varieties, algebraic groups, etc	p. 123
Example: the k-forms of the group SL_n	p. 125
Fields of dimension ≤ 1	p. 128
Linear groups: summary of known results	p. 128
Vanishing of H¹ for connected linear groups	p. 130
Steinberg's theorem	p. 132
Rational points on homogeneous spaces	p. 134
Fields of dimension ≤ 2	p. 139
Conjecture II	p. 139
Examples	p. 140
Finiteness theorems	p. 142
Condition (F)	p. 142
Fields of type (F)	p. 143
Finiteness of the cohomology of linear groups	p. 144
Finiteness of orbits	p. 146
The case k = R	p. 147
Algebraic number fields (Borel's theorem)	p. 149
A counter-example to the "Hasse principle"	p. 149
Bibliographic remarks for Chapter III	p. 154
Regular elements of semisimple groups	p. 155
Introduction and statement of results	p. 155
Some recollections	p. 158
Some characterizations of regular elements	p. 160
The existence of regular unipotent elements	p. 163
Irregular elements	p. 166
Class functions and the variety of regular classes	p. 168
Structure of N	p. 172
Proof of 1.4 and 1.5	p. 176
Rationality of N	p. 178
Some cohomological applications184
Added in proof	p. 185
Complements on Galois cohomology	p. 187
Notation	p. 187
The orthogonal case	p. 188
Applications and examples	p. 189
Injectivity problems	p. 192
The trace form	p. 193
Bayer-Lenstra theory: self-dual normal bases	p. 194
Negligible cohomology classes	p. 196
Bibliography	p. 199
Index	p. 209
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Galois Cohomology

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