Auslander condition and Gorenstein rings

Let $R$ be a left Noetherian ring satisfying the Auslander condition. It is proven that a left $R$-module $M$ is Gorenstein injective if and only if $M$ is strongly cotorsion and $\mathrm{Ext}_R^{i\geq1}(I,M)=0$ for any injective left $R$-module $I$; a right $R$-module $M$ is Gorenstein flat if and only if $M$ is strongly torsion-free and Tor$^R_{i\geq1}(M,J)=0$ for any injective left $R$-module $J$. We also prove that if $R$ is a commutative Noetherian ring with splf $R$ finite, then the local ring $R_{\mathfrak{p}}$ is Gorenstein for every prime ideal $\mathfrak{p}$ of $R$ if and only if the cycles of every acyclic complex of PGF-modules are PGF-modules if and only if every complex of PGF-modules is a dg-PGF complex.