An Auslander-Buchsbaum formula for higher Auslander algebras and applications

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.

19 pages, comments are welcome! It contains one more example and a new appendix explaining how to use QPA to test whether a given higher Auslander algebra is QF-1. More details were added in the introduction