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

TL;DR

The paper characterizes when a Nakayama algebra is minimal Auslander-Gorenstein by a criterion expressed entirely in terms of Ringel's resolution quiver and the parity of the selfinjective dimension. It develops the syzygy filtered algebra $\varepsilon(A)$ and proves an inductive reduction that transfers the problem from $A$ to $\varepsilon(A)$, enabling explicit conditions on leaf distances, predecessor counts, quiver connectivity, and blackness of cyclic vertices. The main results give precise necessary-and-sufficient conditions for odd and even selfinjective dimensions, and the work is extended to applications involving quiver transformations, higher Auslander algebras, and explicit constructions via $\varepsilon^{-1}$. These findings provide a concrete, quiver-based framework for classifying minimal Auslander-Gorenstein Nakayama algebras and their higher analogues.

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 $A$.

Figures (1)

• Figure 1: The quiver $\Delta_n$.

Theorems & Definitions (43)

• Theorem A
• Theorem B
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Corollary 3.2
• Lemma 3.3