Gorenstein modules and dimension over large families of infinite 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.