An Auslander-Buchsbaum formula for higher Auslander algebras and applications

TL;DR

This paper generalizes the Auslander-Buchsbaum formula to higher Auslander algebras, classifies certain algebra classes, and explores their properties and relationships in higher homological dimensions.

Contribution

It introduces a non-commutative version of the Auslander-Buchsbaum formula for higher Auslander algebras and classifies tilted Auslander and QF-1 algebras.

Findings

Classified tilted Auslander algebras and QF-1 algebras of global dimension at most 2.

Established a non-commutative Auslander-Buchsbaum formula for higher Auslander algebras.

Provided a local characterization of higher QF-1 Auslander algebras.

Abstract

We provide a new non-commutative generalisation of the Auslander-Buchsbaum formula for higher Auslander algebras and use this to show that the class of tilted Auslander algebras, studied recently by Zito, and QF-1 algebras of global dimension at most 2, studied by Ringel in the 1970s, coincide. We furthermore give an explicit classification of this class of algebras and present generalisations to higher homological dimensions with a new local characterisation of QF-1 higher Auslander algebras.