Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras

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.

19 pages

References in corpus (6)

• Gorenstein Homological Algebra of Artin Algebras
• Special tilting modules for algebras with positive dominant dimension
• Dominant dimension and tilting modules
• Finitistic Auslander algebras
• Auslander-Gorenstein algebras, standardly stratified algebras and dominant dimensions
• Auslander-Gorenstein algebras from Serre-formal algebras via replication