Gorenstein modules and dimension over large families of infinite groups

TL;DR

This paper characterizes Gorenstein modules over group algebras for large classes of infinite groups, establishing new relationships and properties of these modules in the context of Gorenstein homological algebra.

Contribution

It provides new characterizations of Gorenstein modules over group algebras, extending Gorenstein homological theory to infinite groups and related algebraic structures.

Findings

Weak Gorenstein modules are equivalent to Gorenstein modules in this context

Over rings with finite Gorenstein weak global dimension, Gorenstein projective modules are Gorenstein flat

Determined Gorenstein homological dimensions for specific classes of groups

Abstract

We give characterizations of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over the group algebra for large families of infinite groups and show that every weak Gorenstein projective, weak Gorenstein flat and weak Gorenstein injective module is Gorenstein projective, Gorenstein flat and Gorenstein injective, respectively. These characterizations provide Gorenstein analogues of Benson's cofibrant modules. We deduce that, over a commutative ring of finite Gorenstein weak global dimension, every Gorenstein projective module is Gorenstein flat. Moreover, we study cases where the tensor product and the group of homomorphisms between modules over the group algebra is a Gorenstein module. Finally, we determine the Gorenstein homological dimension of an $\textsc{\textbf{lh}}\mathfrak{F}$-group over a commutative ring of finite Gorenstein weak global dimension.