Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marco Boggi

TL;DR

This work extends Lannes' and Quillen's frameworks to arbitrary profinite groups by developing a topological theory of products and coproducts for families of torsion modules parameterized by profinite spaces, expressed via proÉtalé spaces and Pontryagin duality. The authors construct a precise Lannes–Quillen isomorphism in the profinite setting, linking $T_VH^ullet(G)$ to cohomology of centralizers through étalé-coefficient systems and orbit data, and demonstrate applications to conjugacy separability of $p$-elements and finite $p$-subgroups. The introduced formalism unifies sheaf-theoretic and duality approaches for profinite and discrete coefficients, establishing equivalences and adjunctions that enable robust control of cohomology and subgroup structures in profinite groups. Overall, the paper provides a full profinite analogue of Lannes–Quillen decompositions with concrete consequences for conjugacy separability and subgroup classifications in profinite contexts.

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.

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

TL;DR

This work extends Lannes' and Quillen's frameworks to arbitrary profinite groups by developing a topological theory of products and coproducts for families of torsion modules parameterized by profinite spaces, expressed via proÉtalé spaces and Pontryagin duality. The authors construct a precise Lannes–Quillen isomorphism in the profinite setting, linking to cohomology of centralizers through étalé-coefficient systems and orbit data, and demonstrate applications to conjugacy separability of -elements and finite -subgroups. The introduced formalism unifies sheaf-theoretic and duality approaches for profinite and discrete coefficients, establishing equivalences and adjunctions that enable robust control of cohomology and subgroup structures in profinite groups. Overall, the paper provides a full profinite analogue of Lannes–Quillen decompositions with concrete consequences for conjugacy separability and subgroup classifications in profinite contexts.

Abstract

The Lannes-Quillen theorem relates the mod- cohomology of a finite group with the mod- cohomology of centralizers of abelian elementary -subgroups of , for a prime number. This theorem was extended to profinite groups whose mod- 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 -torsion elements and finite -subgroups.
Paper Structure (38 sections, 46 theorems, 90 equations)

This paper contains 38 sections, 46 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Lannes' $T$-functor
  3. The reduced $T$-functor
  4. Lannes' generalization of Quillen's theorem
  5. A generalization to arbitrary profinite groups
  6. Applications to conjugacy separability
  7. Sheaves and étalé spaces of profinite and discrete torsion modules
  8. Sheaves of profinite and discrete torsion modules over a profinite space
  9. Pontryagin duality for sheaves of profinite and discrete torsion modules
  10. The category of proétalé spaces of profinite $\Rho$-modules over $T$
  11. The category of proétalé spaces of profinite $\Rho$-modules
  12. The category of étalé spaces of discrete $\Rho$-modules
  13. Strongly filtered compactly generated weakly Hausdorff spaces
  14. Pontryagin duality for étalé spaces
  15. (Pro)étalé spaces of finite $\Rho$-modules
  16. ...and 23 more sections

Key Result

Theorem 1.1

For a finite group $G$ and an elementary abelian $p$-group $V$, the map Lannesiso is an isomorphism of unstable algebras over the mod-$p$ Steenrod algebra.

Theorems & Definitions (103)

  • Theorem 1.1: Lannes
  • Definition 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 93 more