Gorenstein homological dimensions and Auslander categories

Abstract

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

15 pages