# 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.