Gorenstein homological dimension and some invariants of groups

TL;DR

This paper introduces and analyzes the Gorenstein homological dimension for groups, establishing its finiteness criteria, exploring its properties under group operations, and comparing it with other algebraic invariants.

Contribution

It provides a new characterization of Gorenstein homological dimension for groups and investigates its behavior under various group-theoretic constructions and relations with other invariants.

Findings

Gorenstein homological dimension is finite iff all RG-modules have finite Gorenstein flat dimension.

Established criteria involving R-pure RG-monic A for Gorenstein homological dimension.

Compared Gorenstein invariants with classical homological dimensions and provided conditions for Gorenstein projective-flat problems.

Abstract

For any group $G$, the Gorenstein homological dimension ${\rm Ghd}_RG$ is defined to be the Gorenstein flat dimension of the coefficient ring $R$, which is considered as an $RG$-module with trivial group action. We prove that ${\rm Ghd}_RG < \infty$ if and only if the Gorenstein flat dimension of any $RG$-module is finite, if and only if there exists an $R$-pure $RG$-monic $R\rightarrow A$ with $A$ being $R$-flat and ${\rm Ghd}_RG = {\rm fd}_{RG}A$, where $R$ is a commutative ring with finite Gorenstein weak global dimension. As applications, properties of ${\rm Ghd}$ on subgroup, quotient group, extension of groups as well as Weyl group are investigated. Moreover, we compare the relations between some invariants such as ${\rm sfli}RG$, ${\rm silf}RG$, ${\rm spli}RG$, ${\rm silp}RG$, and Gorenstein projective, Gorenstein flat and PGF dimensions of $RG$-modules; a sufficient condition for Gorenstein projective-flat problem over group rings is given.