Gorenstein homological dimensions and Auslander categories

In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring $R$. For an $R$-module $M$, we denote by ${\mathrm{G}\mathrm{p}\mathrm{d}}_{R}M$ and ${\mathrm{G}\mathrm{f}\mathrm{d}}_{R}M$ the Gorenstein projective and flat dimensions of $M$, respectively. We show that ${\mathrm{G}\mathrm{p}\mathrm{d}}_{R}M<\mathrm{\infty }$ if and only if ${\rm Gfd}_RM<\infty$providedtheKrulldimensionof$R$ is finite. Moreover, in the case that $R$ is local, we correspond to a dualizing complex $\mathbf{D}$ of $\widehat{R}$, the classes $A^{\mathrm{\prime }}(R)$ and $B^{\mathrm{\prime }}(R)$ of $R$-modules. For a module $M$ over a local ring $R$, we show that $M\in A^{\mathrm{\prime }}(R)$ if and only if ${\rm Gpd}_RM<\infty$orequivalently${\rm Gfd}_RM<\infty$. In dual situation by using the class $B^{\mathrm{\prime }}(R)$, we provide a characterization of Gorenstein injective modules.