Galois Cohomology

ISBN-13: 9783642638664 / Angielski / Miękka / 2012 / 212 str.

Jean-Pierre Serre; P. Ion

Opis

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 . . . . . . . . . . . . . .

Spis treści

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 Poincaré 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-Poincaré characteristic (elementary case).- 5.5 Unramified cohomology.- 5.6 The Galois group of the maximal p-extension of k.- 5.7 Euler-Poincaré 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”.

O autorze

Professor Jean-Pierre Serre ist ein renommierter französischer Mathematiker am College de France in Paris.