$\boldsymbol{τ}$-Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors. (9th July 2017) - Demonet, Laurent; Iyama, Osamu; Jasso, Gustavo
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, that is, 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 every torsion class in $\mathsf{mod}\, A$ is functorially finite. Moreover 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. 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 $Δ(A)$ .