A characterisation of $\tau$-tilting finite algebras

TL;DR

This paper characterizes finite dimensional algebras that are $ au$-tilting finite, establishing a connection with silting modules and ring epimorphisms, and providing criteria for finiteness based on these structures.

Contribution

It provides a complete characterization of $ au$-tilting finite algebras through silting modules and ring epimorphisms, linking algebraic properties to module-theoretic and categorical structures.

Findings

A finite dimensional algebra is $ au$-tilting finite iff it admits no large silting modules.

There is a bijection between support $ au$-tilting modules and certain ring epimorphisms.

Finiteness of $ au$-tilting modules corresponds to finitely many equivalence classes of ring epimorphisms.

Abstract

We prove that a finite dimensional algebra is $\tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $\tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $\tau$-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 $\tau$-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.