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