Projectively coresolved Gorenstein flat dimension of groups

TL;DR

This paper introduces the projectively coresolved Gorenstein flat dimension for groups over rings, establishing its properties, estimating global dimensions, and analyzing special group cases with implications for Gorenstein modules.

Contribution

It defines a new homological dimension for groups, explores its properties, and relates it to existing dimensions and module categories, especially for specific group classes.

Findings

The new dimension shares properties with cohomological and Gorenstein cohomological dimensions.

Provides estimations for the Gorenstein global dimension of group rings.

Shows that for certain groups, Gorenstein projective modules are Gorenstein flat when the ring's global dimension is finite.

Abstract

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group $G$ over a commutative ring $R$ and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of $RG$ in terms of this dimension and the Gorenstein global dimension of $R$. Moreover, we study special cases of groups, such as $\textsc{\textbf{lh}}\mathfrak{F}$-groups, and show that for such a group every Gorenstein projective $RG$-module is Gorenstein flat when the global dimension of $R$ is finite.