Three Results Concerning Auslander Algebras

TL;DR

This paper presents three key findings about Auslander algebras, including a new characterization, a criterion for being glued, and the existence of a special tilting module with hereditary properties.

Contribution

It introduces a novel characterization of Auslander algebras, links glued property to representation-finiteness, and identifies a unique hereditary tilting module within their module categories.

Findings

New characterization of Auslander algebras via hereditary torsion pairs

Glued Auslander algebras are exactly the representation-finite ones

Existence of a hereditary tilting module in any Auslander algebra

Abstract

Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra $\Lambda$ is left or right glued if and only if $\Lambda$ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated.