Description
Springer Galois Cohomology by Jean-Pierre Serre, P. Ion
This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography._x000D_ Table of contents :- _x000D_
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"._x000D_