Foreword
Chapter I. Cohomology of proflnite groups
1. Proflnite groups
1.1 Definition
1.2 Subgroups
1.3 Indices
1.4 Pro-p-groups and Sylow p-subgroups
1.5 Pro-p-groups
2. Cohomology
2.1 Discrete G-modules
2.2 Cochains, cocycles, cohomology
2.3 Low dimensions
2.4 Functoriality
2.5 Induced modules
2.6 Complements
3. Cohomological dimension
3.1 p-cohomological dimension
3.2 Strict cohomological dimension
3.3 Cohomological dimension of subgroups and extensions
3.4 Characterization of the profinite groups G such that cdpG 1
3.5 Dualizing modules
4. Cohomology of pro-p-groups
4.1 Simple modules
4.2 Interpretation of H1: generators
4.3 Interpretation of H2: relations
4.4 A theorem of Shafarevich
4.5 Poincare groups
5. Nonabelian cohomology
5.1 Definition of H~ and of H1
5.2 Principal homogeneous spaces over A - a new definition of
H1G,A
5.3 Twisting
5.4 The cohomology exact sequence associated to a subgroup
5.5 Cohomology exact sequence associated to a normal subgroup
5.6 The case of an abelian normal subgroup
5.7 The case of a central subgroup
5.8 Complements
5.9 A property of groups with cohomological dimension _ 1
Bibliographic remarks for Chapter I
Appendix 1. J. Tate - Some duality theorems
Appendix 2. The Golod-Shafarevich inequality
1. The statement
2. Proof
Chapter II. Gaiois cohomology, the commutative case
1. Generalities
1.1 Galois cohomology
1.2 First examples
2. Criteria for cohomological dimension
2.1 An auxiliary result
2.2 Case when p is equal to the characteristic
2.3 Case when p differs from the characteristic
3. Fields of dimension _1
3.1 Definition
3.2 Relation with the property C1
3.3 Examples of fields of dimension _ 1
4. Transition theorems
4.1 Algebraic extensions
4.2 Transcendental extensions
4.3 Local fields
4.4 Cohomological dimension of the Galois group of an algebraic
number field
4.5 Property Cr
5. p-adic fields
5.1 Summary of known results
5.2 Cohomology of finite Gk-modulea
5.3 First applications
5.4 The Euler-Poincare characteristic elementary case
5.5 Unramified cohomology
5.6 The Galois group of the maximal p-extension of k
5.7 Euler-Poincar6 characteristics
5.8 Groups of multiplicative type
6. Algebraic number fields
6.1 Finite modules - definition of the groups Ptk, A
6.2 The finiteness theorem
6.3 Statements of the theorems of Poitou and ~te
Bibliographic remarks for Chapter II
Appendix. Gaiols cohomology of purely transcendental extensions
1. An exact sequence
2. The local case
3. Algebraic curves and function fields in one variable
4. The case K = kT
5. Notation
6. Killing by base change
7. Manin conditions, weak approximation
and Schinzel''s hypothesis
8. Sieve bounds
Chapter III. Nonabelian Galols cohomology
1. Forms
1.1 Tensors
1.2 Examples
1.3 Varieties, algebraic groups, etc
1.4 Example: the k-forms of the group SLn
2. Fields of dimension _ 1
2.1 Linear groups: summary of known results
2.2 Vanishing of H1 for connected linear groups
2.3 Steinberg''s theorem
2.4 Rational points on homogeneous spaces
3. Fields of dimension _ 2
3.1 Conjecture II
3.2 Examples
4. Finiteness theorems
4.1 Condition F
4.2 Fields of type F
4.3 Finiteness of the cohomology of linear groups
4.4 Finiteness of orbits
4.5 The case k = R
4.6 Algebraic number fields Borel''s theorem
4.? A counter-example to the "Hasse principle"
Bibliographic remarks for Chapter III
Appendix 1. Regular elements of semisimple groups by R. Steinberg
1. Introduction and statement of results
2. Some recollections
3. Some characterizations of regular elements
4. The existence of regular unipotent elements
5. Irregular elements
6. Class functions and the variety of regular classes
7. Structure of N
8. Proof of 1.4 and 1.5
9. Rationality of N
10. Some cohomological applications
11. Added in proof
Appendix 2. Complements on Galois cohomology
1. Notation
2. The orthogonal case
3. Applications and examples
4. Injectivity problems
5. The trace form
6. Bayer-Lenstra theory: self-dual normal bases
7. Negligible cohomology classes
Bibliography
Index
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