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

TL;DR

This paper extends the Lannes-Quillen theorem to all profinite groups, linking their mod-$p$ cohomology with the cohomology of centralizers of abelian elementary $p$-subgroups, and develops new product theories for profinite modules.

Contribution

It formulates and proves a full version of the Lannes-Quillen theorem for all profinite groups, building on and generalizing previous partial results.

Findings

Extended the Lannes-Quillen theorem to all profinite groups.

Developed a theory of products for families of discrete torsion modules.

Applied results to conjugacy separability of $p$-torsion elements.

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 space, 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.