Year: 2026
Type: preprint
Abstract: We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type $FP_\infty$ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality $\mathrm{Gcd}_kG=\mathrm{Ghd}_kG$ for every group $G$ of type $FP_\infty$ over a commutative coefficient ring $k$. For non-locally-finite groups in Kropholler's class ${\scriptstyle\mathbf{LH}}\mathfrak F$, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group $G$ determines its Gorenstein (co)homological dimension, even when $G$ has torsion: We show that $\mathrm{Ghd}_\mathbb{Z}G =h(G)$ and, if $G$ is countable, $h(G)\leq\mathrm{Gcd}_\mathbb{Z}G\leq h(G)+1$. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.
Source: arXiv (Cornell University)
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204176910