Decompositions of the stable module $\infty$-category

TL;DR

This paper explores how the stable module $ty$-category of a finite group G can be decomposed into limits over subgroups, introducing new decompositions and extending existing ones with broader applicability.

Contribution

It introduces three new types of decompositions (centraliser, normaliser, subgroup) for the stable module ty-category and extends the subgroup decomposition to more collections of subgroups.

Findings

Constructed centraliser and normaliser decompositions.

Extended the subgroup decomposition to broader collections.

Showed the extension depends only on the S-equivariant homotopy type.

Abstract

We show that the stable module $\infty$-category of a finite group $G$ decomposes in three different ways as a limit of the stable module $\infty$-categories of certain subgroups of $G$. Analogously to Dwyer's terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The key step in the proof is extending the stable module $\infty$-category to be defined for any $G$-space, then showing that this extension only depends on the $S$-equivariant homotopy type of a $G$-space. The methods used are not specific to the stable module $\infty$-category, so may also be applicable in other settings where an $\infty$-category depends functorially on $G$.