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

TL;DR

This paper explores the structure of $ au$-tilting finite algebras, revealing their geometric and combinatorial properties, including a sphere-like simplicial complex and a bijection with certain bricks, advancing understanding of their module categories.

Contribution

It characterizes $ au$-tilting finite algebras via torsion classes, describes the geometric structure of associated $g$-vector cones, and establishes a bijection with specific bricks, providing new insights into their module theory.

Findings

$ au$-tilting finite algebras have functorially finite torsion classes.

The simplicial complex $ riangle(A)$ is homeomorphic to an $(n-1)$-sphere.

A bijection exists between indecomposable $ au$-rigid modules and certain bricks.

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 $\mod A$ 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.