Gorenstein homological dimension of group extensions

Abstract

We study the Gorenstein homological dimension $\mathrm{Ghd}_k G$ of groups $G$ which are of type $\mathrm{FP}_{\infty}$ over a commutative ring $k$. Our main technical result shows that, when the Gorenstein weak global dimension of a ring $R$ is finite, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $\mathrm{FP}_\infty$ over $k$ with $\mathrm{sfli} k<\infty$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel$&#39;$dman&#39;s theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $\mathrm{FP}_\infty$ over $\mathbb{Z}$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $\mathrm{FP}_\infty$ over a principal ideal domain is realized after passing to a suitable field.