$\mathfrak{X}$-Gorenstein projective dimensions

TL;DR

This paper explores the properties of $rak{X}$-Gorenstein projective dimensions of modules and rings, extending Auslander's theorem to a broader Gorenstein homological context.

Contribution

It introduces new properties of $rak{X}$-Gorenstein projective dimensions and proves a key equality relating global dimension to module dimensions, extending classical theorems.

Findings

$rak{X}$-Gorenstein global dimension equals the supremum of module dimensions.

Properties of $rak{X}$-Gorenstein projective dimensions are established.

Extension of Auslander's theorem to Gorenstein homological setting.

Abstract

In this paper, we mainly investigate the $\mathfrak{X}$-Gorenstein projective dimension of modules and the (left) $\mathfrak{X}$-Gorenstein global dimension of rings. Some properties of $\mathfrak{X}$-Gorenstein projective dimensions are obtained. Furthermore, we prove that the (left) $\mathfrak{X}$-Gorenstein global dimension of ring $R$ is equal to the supremum of the set of $\mathfrak{X}$-Gorenstein projective dimensions of all cyclic (left) $R$-modules. This result extends the well-known Auslander's theorem on the global dimension and its Gorenstein homological version.