When do Foxby classes coincide with the classes of modules of finite Gorenstein dimensions?

Abstract

The relation between the Auslander (resp., Bass) class and the class of modules with finite Gorenstein projective (resp., injective) dimension is well known when these mentioned classes are built with a dualizing module over Noetherian $n$-perfect rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this article we try to extend and sometimes improve some of these results by weakening the condition of being dualizing. Among other results, we prove that a Wakamatsu tilting module with some extra conditions is precisely a module ${}_{R}C$ such that the Bass class ${\mathcal{B}}_C\left(R\right)$ coincides with the class of modules of finite Gorenstein injective dimension.