$\mathcal{T}_C $ -Gorenstein projective, $\mathcal{L}_C $ -Gorenstein injective and $\mathcal{H}_C $ -Gorenstein flat modules

Abstract

The authors introduce and investigate the \(\mathcal{T}_C \)-Gorenstein projective, \(\mathcal{L}_C \)-Gorenstein injective and \(\mathcal{H}_C \)-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the \(\mathcal{T}_C \)-Gorenstein projective or the \(\mathcal{H}_C \)-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the \(\mathcal{L}_C \)-Gorenstein injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the \(\mathcal{T}_C \)(R)-projective and \(\mathcal{L}_C \)(R)-injective dimensions and \(\mathcal{T}_C \)(R)-precovers and \(\mathcal{L}_C \)(R)-preenvelopes. Finally, the authors study the \(\mathcal{H}_C \)-Gorenstein flat modules and extend the Foxby equivalences.