A new characterization of Auslander algebras

TL;DR

This paper provides a new characterization of Auslander algebras by relating the projective dimensions of modules to those of their socles, with implications for identifying Auslander algebras based on global dimension and socle properties.

Contribution

It introduces a novel criterion for Auslander algebras involving projective dimensions of modules and their socles, extending existing characterizations.

Findings

Projective dimension of a module is at most one iff its socle's projective dimension is at most one.

A finite dimensional algebra is an Auslander algebra if its global dimension is at most two and certain socle conditions hold.

Injective modules are projective iff their socles have projective dimension at most one under specific conditions.

Abstract

Let $\Lambda$ be a finite dimensional Auslander algebra. For a $\Lambda$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an application, we give a new characterization of Auslander algebra $\Lambda$, and prove that a finite dimensional algebra $\Lambda$ is an Auslander algebra provided its global dimension gl.d\,$\Lambda\leq2$ and an injective $\Lambda$-module is projective if and only if the projective dimension of its socle is at most one.