Defect Invariant Nakayama Algebras

TL;DR

This paper explores the structure of cyclic Nakayama algebras, identifying those with invariant defects and classifying specific minimal Auslander-Gorenstein and dominant Auslander-regular cases of global dimension three, using cluster-tilting theory.

Contribution

It introduces the concept of defect invariant Nakayama algebras, classifies certain minimal Auslander-Gorenstein and dominant Auslander-regular algebras, and constructs cluster-tilting objects via the Auslander-Iyama correspondence.

Findings

Existence of countably many cyclic Nakayama algebras with syzygy filtered algebra isomorphic to a given algebra.

Identification of a unique algebra with invariant defects.

Classification of minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three.

Abstract

We show that for a given Nakayama algebra $\Theta$, there exist countably many cyclic Nakayama algebras $\Lambda_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $\Lambda_i$ is isomorphic to $\Theta$ and we describe those algebras $\Lambda_i$. We show, among these algebras, there exists a unique algebra $\Lambda$ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both $\Theta$ and $\Lambda$. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra.