On selfinjective algebras of stable dimension zero

TL;DR

This paper investigates the stable dimension of selfinjective algebras over algebraically closed fields, establishing that a stable dimension of zero implies the algebra is representation-finite.

Contribution

It proves that selfinjective algebras with stable dimension zero are necessarily representation-finite, linking stable dimension to representation type.

Findings

Stable dimension zero implies representation-finiteness.

Established a criterion connecting stable dimension and algebra classification.

Enhanced understanding of the structure of selfinjective algebras.

Abstract

Let $A$ be a selfinjective algebra over an algebraically closed field. We study the stable dimension of $A$, which is the dimension of the stable module category of $A$ in the sense of Rouquier. Then we prove that $A$ is representation-finite if the stable dimension of $A$ is $0$.