A characterisation of $τ$-tilting finite algebras

Abstract

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 $A\longrightarrow B$ with ${\rm Tor}_1^A(B,B)=0$. 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.