Relative Gorenstein Dimensions

Abstract

In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of GC-projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of GC-injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of GC-projective modules and the Bass class associated to C on one side, and the class of G\({_{C^{\vee}}}\) -injective modules (C∨ = HomR (C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other.