A characterisation of τ-tilting finite algebras

We prove that a finite dimensional algebra is τ-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a τ- tilting finite algebra A there is a bijection between isomorphism classes of basic support τ-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms A −→ B with TorA1 (B, B) = 0. It follows that a finite dimensional algebra is τ-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.