Finiteness criteria for Gorenstein homological dimension and some invariants of groups

TL;DR

The article develops finiteness criteria for the Gorenstein homological dimension of groups over rings with finite Gorenstein weak global dimension by introducing weak characteristic modules and establishing equivalences with PGF- and Gorenstein flat-structure conditions. It connects Ghd$_kG$ to Gcd$_kG$ and to generalized homological/cohomological dimensions, yielding Serre-type results and duality phenomena via Pontryagin duals. The work applies these ideas to infinite groups (type $\Phi_k$ and LH$\mathfrak{F}$FP$_\infty$) to obtain Gorenstein analogues of classical properties and shows how invariants like sfli and Gwgl.dim behave under extensions and finite-index subgroups. Overall, it provides a unified framework for relating Gorenstein homological and cohomological dimensions, along with practical bounds and subadditivity results for group rings $kG$.

Abstract

In this paper, we study finiteness criteria for the Gorenstein homological dimension of groups over a commutative ring of finite Gorenstein weak global dimension and provide estimates for the Gorenstein weak global dimension of group rings. As a result, we obtain Gorenstein analogues of well known properties in classical homological algebra over large families of infinite groups. Moreover, we prove that over a commutative ring of finite Gorenstein weak global dimension, the Gorenstein cohomological dimension of a group G bounds its Gorenstein homological dimension. Finally, we compare the generalized cohomological dimension and the generalized homological dimension of a group.

Theorems & Definitions (104)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof