$τ$-tilting finite algebras, bricks and $g$-vectors

Abstract

The class of support $τ$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $τ$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $τ$-rigid modules. We show that $A$ is $τ$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $τ$-tilting module form a simplicial complex $Δ(A)$. We show that if $A$ is $τ$-tilting finite, then $Δ(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $τ$-tilting modules can be recovered from the geometry of $Δ(A)$. Finally we give a bijection between indecomposable $τ$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $τ$-tilting finite algebras.