2015/03/31 by Laurent Demonet, Osamu Iyama, Gustavo Jasso · 1 citation
Mathematics · #math.RT #msc:18E40 #msc:16G20
paper · pdf · doi:10.1093/imrn/rnx135
published as International Mathematics Research Notices IMRN, Volume 2019, Issue 3, 7 February 2019, Pages 852-892 · 29 pages. Changed title. Added Theorem 6.5 and Proposition 6.6
arxiv created 2017/06/20 · arxiv updated 2019/02/13
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.