Auslander conditions and tilting-like cotorsion pairs

TL;DR

This paper investigates modules satisfying the Auslander condition, establishing their homological properties, and explores their relation to Gorenstein rings and tilting-like cotorsion pairs, providing criteria for the Auslander and Reiten conjecture.

Contribution

It characterizes the homological behavior of Auslander condition modules and links them to Gorenstein rings and tilting cotorsion pairs, advancing understanding of module categories.

Findings

Cycle modules with Auslander condition stay within the class

Resolving subcategory condition equivalent to Gorenstein property

Criteria for the Auslander and Reiten conjecture validity

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.