$\boldsymbol{\tau}$-Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors. (9th July 2017)

• Record Type: Journal Article
• Authors: Demonet, Laurent; Iyama, Osamu; Jasso, Gustavo
• Is Part Of: International mathematics research notices. Volume 2019:Number 3(2019)
• Page Start: 852
• Page End: 892
• Publication Date: 2017-07-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx135
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: The class of support $\tau$ -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 $\tau$ -tilting finite algebras, that is, finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\tau$ -rigid modules. We show that $A$ is $\tau$ -tilting finite if and only if every torsion class in $\mathsf{mod}\, A$ is functorially finite. Moreover we give a bijection between indecomposable $\tau$ -rigid $A$ -modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\tau$ -tilting finite algebras. We observe that cones generated by $g$ -vectors of indecomposable direct summands of each support $\tau$ -tilting module form a simplicial complex $\Delta(A)$ . We show that if $A$ is $\tau$ -tilting finite, then $\Delta(A)$ is homeomorphic to an $(n-1)$ -dimensional sphere, and moreover the partial order on support $\tau$ -tilting modules can be recovered from the geometry of $\Delta(A)$ .