Module Categories for Group Algebras over Commutative 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.