Stratification and duality for homotopical groups

TL;DR

This paper extends classical theorems from finite groups to homotopical groups, establishing stratification, duality, and support theory for module spectra over their cochain algebras, thus broadening the understanding of their algebraic and homotopical properties.

Contribution

It generalizes key theorems from finite groups to homotopical groups and develops a support-theoretic classification for module spectra over their cochain algebras.

Findings

Category of module spectra is stratified and costratified for various p-local groups.

Established a homotopical Gorenstein duality for p-compact groups.

Extended classical theorems like Quillen's to a homotopical setting.

Abstract

We generalize Quillen's $F$-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over $C^*(B\mathcal{G},\mathbb{F}_p)$ is stratified and costratified for a large class of $p$-local compact groups $\mathcal{G}$ including compact Lie groups, connected $p$-compact groups, and $p$-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that $p$-compact groups admit a homotopical form of Gorenstein duality.