Retractions and Gorenstein homological properties

TL;DR

This paper investigates how left retractions of algebras influence Gorenstein homological properties, providing classifications and linking Nakayama algebras to self-injective cases through singularity categories.

Contribution

It introduces a new approach to study Gorenstein properties via left retractions and classifies small Nakayama algebras based on these properties.

Findings

Existence of a sequence of left retractions linking Nakayama algebras to self-injective algebras.

Singularity categories of Nakayama algebras are equivalent to stable categories of certain self-injective algebras.

Classification of small Nakayama algebras by Gorenstein homological properties.

Abstract

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra $A$, there exists a connected self-injective Nakayama algebra $A'$ such that there is a sequence of left retractions linking $A$ to $A'$; in particular, the singularity category of $A$ is triangle equivalent to the stable category of $A'$. We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.