We study homological behavior of modules satisfying the Auslander condition. Assume that $\mathcal{ AC}$ is the class of left $R$-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in $\mathcal{ AC}$ belongs to $\mathcal{ AC}$ for any ring $R$. As a consequence, we show that for any left Noetherian ring $R$, $\mathcal{ AC}$ is a resolving subcategory of the category of left $R$-modules if and only if $_RR$ satisfies the Auslander condition if and only if each Gorenstein projective left $R$-module belongs to $\mathcal{ AC}$. As an application, we prove that, for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if $\mathcal{ AC}$ coincides with the class of Gorenstein projective left $R$-modules if and only if $({ \mathcal{ AC}^{ < \infty}},(\mathcal{ AC}^{ <\infty})^\bot)$ is a tilting-like cotorsion pair if and only if (${ \mathcal{ AC}^{ < \infty}},\mathcal{ I}$) is a tilting-like cotorsion pair, where $\mathcal{ AC}^{ <\infty}$ is the class of left $R$-modules with finite $\mathcal{ AC}$-dimension and $\mathcal{ I}$ is the class of injective left $R$-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.
Categories: math.RA math.KT