$\tau$-tilting finite algebras, bricks and $g$-vectors

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, i.e. 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 very torsion class in $\mathrm{mod}A$ is functorially finite. 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 $\Delta (A)$. Finally 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.

29 pages. Changed title. Added Theorem 6.5 and Proposition 6.6

References in corpus (2)

• Torsion classes, wide subcategories and localisations
• A reduction theorem for $\tau$-rigid modules

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