Module categories for group algebras over commutative rings

TL;DR

This paper constructs a new stable module category for finite groups over arbitrary commutative rings, creating a triangulated category with desirable properties for studying module representations.

Contribution

It introduces a localized stable module category over commutative rings, connecting it with homotopy and derived categories through a recollement structure.

Findings

Established a compactly generated triangulated category for kG-modules

Connected stable, homotopy, and derived categories via recollement

Provided a framework for studying modules over group algebras over rings

Abstract

We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.