The resolution quiver of Nakayama algebras which are minimal Auslander-Gorenstein

TL;DR

This paper provides a criterion based on Ringel's resolution quiver and the parity of the selfinjective dimension to determine when a Nakayama algebra is minimal Auslander-Gorenstein.

Contribution

It introduces a new criterion for Nakayama algebras to be minimal Auslander-Gorenstein, utilizing Ringel's resolution quiver and selfinjective dimension parity.

Findings

The criterion effectively characterizes minimal Auslander-Gorenstein Nakayama algebras.

Parity of the selfinjective dimension is crucial in the criterion.

The approach links algebraic properties with combinatorial data from the resolution quiver.

Abstract

Let $A$ be a Nakayama algebra. Using Ringel's resolution quiver, we give a criterion to decide whether $A$ is minimal Auslander-Gorenstein. The criterion strongly relies on the parity of the selfinjective dimension of $A$.