# tau-Tilting Finite Algebras With Non-Empty Left Or Right Parts Are Representation-Finite

$$\tau$$-tilting theory was introduced by Adachi, Iyama and Reiten as a far-reaching generalization of classical tilting theory for finite dimensional associative algebras. One of the main classes of objects in the theory is that of $$\tau$$-rigid modules: a module $$M$$ over an algebra $$\Lambda$$ is $$\tau$$-rigid if $$\operatorname{Hom}_{\Lambda}(M,\tau M)=0$$, where $$\tau M$$ denotes the Auslander-Reiten translation of $$M$$; such a module $$M$$ is called $$\tau$$-tilting if the number $$|M|$$ of non-isomorphic indecomposable summands of $$M$$ equals the number of isomorphism classes of simple $$\Lambda$$-modules. Recently, a new class of algebras was introduced by Demonet, Iyama, Jasso called $$\tau$$-tilting finite algebras. They are defined as finite dimensional algebras with only a finite number of isomorphism classes of basic $$\tau$$-tilting modules.

An obvious sufficient condition to be $$\tau$$-tilting finite is to be representation-finite. In general, this condition is not necessary. The aim of this talk is to show for algebras $$\Lambda$$ such that $$L_\Lambda$$ or $$R_\Lambda\neq\emptyset$$ , representation-finiteness and $$\tau$$-tilting finiteness are equivalent conditions.