Galois Cohomology [Hardcover]

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.

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 . . . . . . . . . . . . . .I. Cohomology of profinite groups.- ?1. Profinite groups.- 1.1 Definition.- 1.2 Subgroups.- 1.3 Indices.- 1.4 Pro-p-groups and SylsŠ