Cohomology of p-adic Analytic Groups

Abstract

The purpose of this article is to give an exposition on the cohomology of compact p-adic analytic groups. The cohomology theory of profinite groups was initiated by J. Tate and developed by J-P. Serre [23] in the sixties, with applications to number theory. In his extraordinary work on p-adic analytic groups [17], M. Lazard also considered their cohomology and proved two striking theorems: Lazard’s first theorem states that a compact p-adic analytic group G is a virtual Poincare duality group; his second theorem states that the rational cohomology of G coincides with the G-stable cohomology of its associated \( {\mathbb{Q}_P}\) -Lie algebra L (G). Our main goal is to discuss these results of Lazard in the spirit of the treatment of the structure of p-adic analytic groups in [10]. We also wish to emphasize the close parallels with the theory of discrete duality groups. In order to achieve this goal we need to set up the appropriate homological algebra.