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Galois Cohomology


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Table of Contents

I. Cohomology of profinite groups.- 1. Profinite 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 Fimctoriality.- 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 cdp(G) ? 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 H0 and of H1.- 5.2 Principal homogeneous spaces over A - a new definition of H1(G,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.- II. Galois 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-modules.- 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-Poincare characteristics.- 5.8 Groups of multiplicative type.- 6. Algebraic number fields.- 6.1 Finite modules - definition of the groups Pi(k, A).- 6.2 The finiteness theorem.- 6.3 Statements of the theorems of Poitou and Tate.- III. Nonabelian Galois 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).- 4.7 A counter-example to the "Hasse principle".

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