A brick version of a theorem of Auslander

TL;DR

This paper establishes a characterization of $ au$-tilting finite algebras via the finiteness of bricks, linking torsion classes and algebra finiteness properties.

Contribution

It provides a new criterion for $ au$-tilting finiteness based on the finiteness of bricks, extending classical results in representation theory.

Findings

Finite dimensional algebra is $ au$-tilting finite iff all bricks are finitely generated.

Existence of proper locally maximal torsion classes in $ au$-tilting infinite algebras.

Characterization of $ au$-tilting finiteness through brick finiteness.

Abstract

We prove that a finite dimensional algebra $\Lambda$ is $\tau-$tilting finite if and only if all the bricks over $\Lambda$ are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for $\tau-$tilting infinite algebras.