On the finiteness of the Gorenstein dimension for Artin algebras

TL;DR

This paper characterizes Artin algebras with finite Gorenstein dimension by showing that non-Gorenstein algebras have indecomposable modules with infinite Gorenstein projective and injective dimensions, generalizing previous global dimension results.

Contribution

It proves that non-Gorenstein Artin algebras have indecomposable modules with infinite Gorenstein dimensions, extending earlier characterizations of finite global dimension.

Findings

Non-Gorenstein algebras have modules with infinite Gorenstein dimensions.

Provides a new characterization of algebras with finite Gorenstein dimension.

Generalizes previous results on global dimension to Gorenstein dimension.

Abstract

In \cite{SSZ}, the authors proved that an Artin algebra $A$ with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterisation of algebras with finite global dimension. We prove in this article that an Artin algebra $A$ that is not Gorenstein has an indecomposable $A$-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterisation of algebras with finite Gorenstein dimension. We show that this gives a proper generalisation of the result in \cite{SSZ} for Artin algebras.