A $\tau$-tilting approach to the first Brauer-Thrall conjecture

TL;DR

This paper investigates modules over finite dimensional algebras with division ring endomorphism algebras, establishing a key equivalence between finiteness of such modules and bounded module length, using $ au$-tilting theory.

Contribution

It introduces a $ au$-tilting framework to characterize when modules with division ring endomorphisms are finite in number, linking module length bounds to module category finiteness.

Findings

Finitely many modules with division ring endomorphisms iff their lengths are bounded

Provides a $ au$-tilting perspective on the first Brauer-Thrall conjecture

Establishes a criterion for module category finiteness based on module length

Abstract

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded.