Gorenstein Projective Dimensions of Modules over Minimal Auslander-Gorenstein Algebras

In this article we investigate the relations between the Gorenstein projective dimensions of Lambda-modules and their socles for n-minimal Auslander-Gorenstein algebras Lambda. 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 if and only if its socle has Gorenstein projective dimension at most n if and only if N is cogenerated by a projective Lambda-module. Furthermore, we show that n-minimal Auslander-Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.