Auslander conditions and tilting-like cotorsion pairs

Abstract

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.