$τ$-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.