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

By a result claimed by Pogorzały 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 Pogorzały. We will analyse the structure of symmetric stably biserial algebras and show that in characteristic $\neq2$ 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.