Gorenstein cohomological dimension and stable categories for groups

TL;DR

This paper investigates the Gorenstein cohomological dimension of groups over various rings, establishing model structures and equivalences among stable categories, and generalizing known results to broader contexts.

Contribution

It introduces a model structure on fibrant modules, relates stable categories to Gorenstein projective modules, and extends results beyond rings of finite global dimension.

Findings

Characterization of finiteness of Gorenstein cohomological dimension.

Equivalence of homotopy and stable categories under certain conditions.

Generalization of results to more general coefficient rings.

Abstract

First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in literature obtained over the coefficient ring $\mathbb{Z}$ or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category $\mathcal{F}ib$ consisting of fibrant $RG$-modules, and show that the homotopy category $\mathrm{Ho}(\mathcal{F}ib)$ is triangle equivalent to both the stable category $\underline{\mathcal{C}of}(RG)$ of Benson's cofibrant modules, and the stable module category ${\rm StMod}(RG)$. The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that ${\rm Gcd}_RG<\infty$, ${\rm Ho}(\mathcal{F}ib)$ is equivalent to the stable category of Gorenstein projective $RG$-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective $RG$-modules.