On Auslander-Type Conditions of Modules

TL;DR

This paper characterizes when a Noetherian ring satisfies the Auslander condition through various module-theoretic conditions, and explores implications for Artinian algebras, Gorenstein, and Auslander-regular rings.

Contribution

It provides new equivalent conditions for the Auslander property in rings and modules, and links these to Gorenstein and Auslander-regular ring characterizations.

Findings

Equivalence of Auslander condition with flat and injective module properties.

Characterization of Gorenstein property via finitely generated modules satisfying the Auslander condition.

New criteria for Auslander-Gorenstein and Auslander-regular rings.

Abstract

We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the $i$th term in a minimal injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the injective envelope (resp. flat cover) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$, and if and only if any of the opposite versions of the above conditions hold true. Furthermore, we prove that for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander-Gorenstein rings and Auslander-regular rings.