An Auslander-Buchsbaum formula for higher Auslander algebras and applications

TL;DR

The paper develops a non-commutative Auslander-Buchsbaum framework for finite-dimensional algebras by replacing depth with dominant dimension in higher Auslander theory, establishing a dual Ext relation $\operatorname{Ext}_A^n(\underline{A}, M) \cong D\operatorname{Ext}_A^{d-n}(M, \overline{A})$ for $A$ with $\operatorname{gldim} A = \operatorname{domdim} A = d$ and any suitable $M$, which yields $\operatorname{pdim} M + \operatorname{domdim} M = d$ and its injective analogue. It connects tilted Auslander algebras to $QF-1$ algebras of global dimension at most $2$, providing explicit classifications (e.g., Dynkin types and endomorphism algebras of projective/injective summands) and a local two-condition characterisation for higher $QF-1$ algebras, accessible via torsion-pair methods. The work also furnishes concrete constructions and tests, including a GAP/QPA implementation to verify the $QF-1$ property for higher Auslander algebras, and introduces families like the Nakayama algebras $A_n$ that realize $QF-1$ higher Auslander algebras across dimensions. Overall, the results deepen the interaction between higher homological algebra, tilting theory, and quasi-Frobenius phenomena, with practical criteria for classifying and constructing these 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.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem A: see Theorems \ref{['domdimprojdimformulas']} and \ref{['thm3dot7']}
• Corollary : \ref{['cor3dot8']}
• Theorem B: see Theorem \ref{['theoremB']}
• Theorem C: see Theorem \ref{['TheoremC']}
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Theorem 3.2