Lannes' T-functor and mod-p cohomology of profinite groups

Abstract

The Lannes-Quillen theorem relates the mod-p cohomology of a finite group G with the mod-p cohomology of centralizers of abelian elementary p-subgroups of G, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-p cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space¹, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of p-torsion elements and finite p-subgroups.