Jean-Pierre Serre. Galois Cohomology. translated from the French by Patrick Ion. Springer

Transcription

1 Jean-Pierre Serre Galois Cohomology translated from the French by Patrick Ion Springer

2 Table of Contents Foreword Chapter I. Cohomology of profinite groups 1. Profinite groups Definition Subgroups Indices Prop-groups and Sylow p-subgroups Pro-p-groups 7 2. Cohomology Discrete G-modules Cochains, cocycles, cohomology Low dimensions Functoriality Induced modules Complements Cohomological dimension p-cohomological dimension Strict cohomological dimension Cohomological dimension of subgroups and extensions Characterization of the profinite groups G such that cd p (G) < Dualizing modules Cohomology of pro-p-groups Simple modules Interpretation of H 1 : generators Interpretation of H 2 : relations A theorem of Shafarevich Poincare groups 38

3 VIII Table of Contents 5. Nonabelian cohomology Definition of H and of H Principal homogeneous spaces over A - a new definition of H l {G,A) Twisting The cohomology exact sequence associated to a subgroup Cohomology exact sequence associated to a normal subgroup The case of an abelian normal subgroup The case of a central subgroup Complements A property of groups with cohomological dimension < 1 57 Bibliographic remarks for Chapter I 60 Appendix 1. J. Tate - Some duality theorems 61 Appendix 2. The Golod-Shafarevich inequality The statement Proof 67 Chapter II. Galois cohomology, the commutative case 1. Generalities Galois cohomology First examples Criteria for cohomological dimension An auxiliary result Case when p is equal to the characteristic Case when p differs from the characteristic Fields of dimension < Definition Relation with the property (Ci) Examples of fields of dimension < Transition theorems Algebraic extensions Transcendental extensions Local fields Cohomological dimension of the Galois group of an algebraic number field Property (C r ) 87

4 Table of Contents IX 5. p-adic fields Summary of known results Cohomology of finite Gfc-modules First-applications The Euler-Poincare characteristic (elementary case) Unramified cohomology The Galois group of the maximal p-extension of A; Euler-Poincare characteristics Groups of multiplicative type Algebraic number fields Finite modules - definition of the groups P l (k, A) The finiteness theorem Statements of the theorems of Poitou and Tate 107 Bibliographic remarks for Chapter II 109 Appendix. Galois cohomology of purely transcendental extensions An exact sequence The local case Ill 3. Algebraic curves and function fields in one variable The case K = k(t) Notation Killing by base change Manin conditions, weak approximation and Schinzel's hypothesis Sieve bounds 117 Chapter III. Nonabelian Galois cohomology 1. Forms Tensors Examples Varieties, algebraic groups, etc Example: thefc-formsof the group SL n Fields of dimension < Linear groups: summary of known results Vanishing of H 1 for connected linear groups Steinberg's theorem Rational points on homogeneous spaces Fields of dimension < Conjecture II Examples 140

5 X Table of Contents 4. Finiteness theorems Condition (F) Fields of type (F) Finiteness of the cohomology of linear groups Finiteness of orbits The case k - R Algebraic number fields (Borel's theorem) A counter-example to the "Hasse principle" 149 Bibliographic remarks for Chapter III 154 Appendix 1. Regular elements of semisimple groups (by R. Steinberg) Introduction and statement of results Some recollections Some characterizations of regular elements The existence of regular unipotent elements Irregular elements Class functions and the variety of regular classes Structure ofat Proof of 1.4 and Rationality of N Some cohomological applications Added in proof 185 Appendix 2. Complements on Galois cohomology Notation The orthogonal case Applications and examples Injectivity problems The trace form Bayer-Lenstra theory: self-dual normal bases Negligible cohomology classes 196 Bibliography 199 Index 209