Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras

TL;DR

This paper explores the relationship between Gorenstein projective dimensions of modules and their socles over minimal n-Auslander-Gorenstein algebras, providing characterizations and descriptions of projective-injective modules.

Contribution

It establishes new links between module Gorenstein projective dimensions and socles, and characterizes minimal n-Auslander-Gorenstein algebras through these relations.

Findings

Modules have Gorenstein projective dimension ≤ n iff their socles do.

Minimal n-Auslander-Gorenstein algebras characterized by module-socle Gorenstein dimensions.

Projective-injective modules described via socle properties.

Abstract

In this article we investigate the relations between the Gorenstein projective dimensions of $\Lambda$-modules and their socles for minimal n-Auslander-Gorenstein algebras $\Lambda$ in the sense of Iyama and Solberg \cite{IS}. First we give a description of projective-injective $\Lambda$-modules in terms of their socles. Then we prove that a $\Lambda$-module $N$ has Gorenstein projective dimension at most n iff its socle has Gorenstein projective dimension at most n iff $N$ is cogenerated by a projective $\Lambda$-module. Furthermore, we show that minimal n-Auslander-Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.