$\tau$-Tilting Finite Algebras With Nonempty Left Or Right Parts Are Representation-Finite

Abstract

Let $\Lambda$ be a finite dimensional algebra such that $\mathcal{L}_{\Lambda}$ or $\mathcal{R}_{\Lambda}\neq\emptyset$. Then $\Lambda$ is $\tau$-tilting finite if and only if $\Lambda$ is representation-finite.