The stable module category and model structures for hierarchically defined groups

TL;DR

This paper constructs a new stable module category for a broad class of infinite groups, providing a model category framework and characterizing compact objects related to Gorenstein dimensions.

Contribution

It introduces a tensor-triangulated stable category for hierarchically defined groups and establishes Quillen equivalences with existing model categories.

Findings

Constructed a compactly generated tensor-triangulated stable category for infinite groups.

Characterized when the trivial module is a compact object in this category.

Established Quillen equivalences with other known model categories.

Abstract

In this work we construct a compactly generated tensor-triangulated stable category for a large class of infinite groups, including those in Kropholler's hierarchy $\mathrm{LH}\mathfrak{F}$. This can be constructed as the homotopy category of a certain model category structure, which we show is Quillen equivalent to several other model categories, including those constructed by Bravo, Gillespie, and Hovey in their work on stable module categories for general rings. We also investigate the compact objects in this category. In particular, we give a characterisation of those groups of finite global Gorenstein AC-projective dimension such that the trivial representation $\mathbb{Z}$ is compact.