Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups

Abstract

1.1 Let G be a profinite group and p be a fixed prime. In this paper we will be concerned with H∗ c (G;Fp), the continuous cohomology of G with coefficients in the trivial module Fp. We will abbreviate H∗ c (G;Fp) by H∗(G;Fp), or simply by H∗G if p is understood from the context. We recall that if G is the (inverse) limit of finite groups Gi then H∗G = colim H∗Gi. Throughout this paper we will assume that H∗G is finitely generated as Fp algebra. By work of Lazard [La] it is known that this holds for many interesting groups, for example for profinite p analytic groups like GL(n,Zp), the general linear groups over the p adic integers. In case H∗G is finitely generated as Fp algebra Quillen has shown [Q1] that there are only finitely many conjugacy classes of elementary abelian p subgroups of G (i.e. groups isomorphic to (Z/p) for some natural number n). In other words, the following category A(G) is equivalent to a finite category: objects of A(G) are all elementary abelian p subgroups of G, and if E1 and E2 are elementary abelian p subgroups of G, then the set of morphisms from E1 to E2 in A(G) consists precisely of those homomorphisms α : E1 −→ E2 of abelian groups for which there exists an element g ∈ G with α(e) = geg−1 ∀e ∈ E1. The category A(G) plays an important role both in Quillen’s results and in the work presented here. This category entered into Quillen’s work as follows. The assignment E 7→ H∗E extends to a functor from the opposite category A(G)op to graded Fp algebras and the restriction homomorphisms H∗G −→ H∗E (for E running through the elementary abelian p subgroups of G) induce a canonical map of algebras q : H∗G −→ limA(G)op H∗E.