τ-Tilting Finite Algebras, Bricks, and 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 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) .