On stably biserial algebras and the Auslander-Reiten conjecture for special biserial algebras

TL;DR

This paper investigates the relationship between stably biserial and special biserial algebras, proving the Auslander-Reiten conjecture for this class and clarifying their structural properties in various characteristics.

Contribution

It provides a detailed proof that selfinjective special biserial algebras are stably biserial and confirms the Auslander-Reiten conjecture for these algebras.

Findings

Selfinjective special biserial algebras are stably biserial.

In characteristic not 2, symmetric special biserial and symmetric stably biserial algebras coincide.

The Auslander-Reiten conjecture holds for special biserial algebras.

Abstract

By a result claimed by Pogorza\l{}y selfinjective special biserial algebras can be stably equivalent only to stably biserial algebras and these two classes coincide. By an example of Ariki, Iijima and Park the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes we provide a detailed proof of the fact that a selfinjective special biserial algebra can be stably equivalent only to a stably biserial algebra following some ideas from the paper by Pogorza\l{}y. We will analyse the structure of symmetric stably biserial algebras and show that in characteristic $\neq 2$ the classes of symmetric special biserial (Brauer graph) algebras and symmetric stably biserial algebras indeed coincide. Also, we provide a proof of the Auslander-Reiten conjecture for special biserial algebras.