Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras

Abstract

In this article we investigate the relations between the Gorenstein projective dimensions of Λ-modules and their socles for minimal n-Auslander-Gorenstein algebras Λ in the sense of Iyama and Solberg [8]. First we give a description of projectiveinjective Λ-modules in terms of their socles. Then we prove that a Λ-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 Λ-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.