Gorenstein analogues of a projectivity criterion over group algebras

TL;DR

The paper develops Gorenstein analogues of Jo's projectivity criterion for group rings over a commutative base $A$, introducing and exploiting invariants such as $\mathrm{Gpd}$, $\mathrm{Gfd}$, $\mathrm{Gid}$ and $\mathrm{Ghd}$, and connecting them to finiteness concepts via the base ring invariants $\mathrm{spli}(A)$ and $\mathrm{sfli}(A)$. By placing the group $G$ in Kropholler's hierarchy with base classes $\mathscr{X}_{\mathrm{Char}}$ and $\mathscr{X}_{\mathrm{WChar}}$, the authors derive three Gorenstein versions of the original question: a projective version (finite $G$ under suitable hypotheses), a flat version yielding $\mathrm{Ghd}_A(G)=0$ and, under extra Noetherian hypotheses, local finiteness; and an injective version establishing equality of injective-related invariants and finiteness in the integral case $A=\mathbb{Z}$. They further introduce the invariant $\kappa_{\mathrm{inj}}(AG;\mathscr{X})$ to relate injective dimensions across subgroups and prove an equality framework for four invariants, yielding structural insights and guiding open questions about which rings ensure these finiteness properties. The results illuminate the interplay between Gorenstein homological properties of group rings and the underlying group structure, and they extend classical results to broader bases, with explicit considerations of when $A=\mathbb{Z}$ is essential.

Abstract

We formulate and answer Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in this article, we deal with more general commutative base rings. We make use of the vast developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we also improve and generalize several existing results from this area along the way.

Theorems & Definitions (45)

• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.7
• Theorem 3.1
• Lemma 3.2