Gorenstein homological dimension and some invariants of groups

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.