On band modules and $\tau$-tilting finiteness

TL;DR

This paper investigates the properties of band modules in finite dimensional algebras, linking their structure to $ au$-tilting finiteness, and provides criteria for specific algebra classes like Brauer graph algebras.

Contribution

It establishes a characterization of $ au$-tilting finiteness for special biserial and Brauer graph algebras via band modules and their endomorphisms.

Findings

A special biserial algebra is $ au$-tilting finite iff no band module is a brick.

Criteria for $ au$-tilting finiteness of Brauer graph algebras based on Brauer graph.

Properties of torsion classes containing band modules are described.

Abstract

In this paper, motivated by a $\tau$-tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $\tau$-tilting finite if and only if no band module is a brick. We also recover a criterion for the $\tau$-tilting finiteness of Brauer graph algebras in terms of the Brauer graph.