New invariants of stable equivalences of algebras

TL;DR

This paper introduces new invariants that are preserved under stable equivalences of Artin algebras, providing insights into homological properties and verifying conjectures for specific algebra classes.

Contribution

It develops novel invariants of stable equivalences and applies them to verify the Auslander--Reiten conjecture for certain algebra classes.

Findings

Stable equivalences preserve homological data related to finitistic dimension.

Stable equivalences induce stable equivalences of Frobenius parts for algebras with positive -dominant dimensions.

Verification of the Auslander--Reiten conjecture for principal centralizer matrix algebras and Frobenius-finite algebras.

Abstract

We show that stable equivalences between Artin algebras without nodes preserve homological data that provide upper bounds for finitistic dimension, and that stable equivalences between Artin algebras with positive $\nu$-dominant dimensions induce stable equivalences of their Frobenius parts. As an application of our new methods developed, we verify the Auslander--Reiten conjecture on stable equivalences for two rather different classes of algebras: principal centralizer matrix algebras over arbitrary fields and Frobenius-finite algebras over algebraically closed fields.