On the existence of certain modules of finite Gorenstein homological dimensions

TL;DR

This paper explores the existence of modules with finite Gorenstein homological dimensions over commutative Noetherian local rings, extending classical results related to Cohen-Macaulay modules.

Contribution

It investigates Gorenstein analogues of known characterizations of Cohen-Macaulay rings via modules with finite injective or projective dimensions.

Findings

Establishes conditions for modules with finite Gorenstein dimensions

Extends classical Cohen-Macaulay characterizations to Gorenstein context

Provides new insights into Gorenstein homological properties

Abstract

Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective dimension. In this paper, we investigate the Gorenstein analogues of these facts.