Semibricks and Brick-finite algebras

Abstract

Let $\Lambda$ be a finite-dimensional algebra. Using the kappa order on the lattice of torsion classes with canonical join representations, we obtain an equivalent condition for $\Lambda$ to be brick-finite. We show that $\Lambda$ is brick-finite if and only if every widely generated torsion class in $\modd \Lambda$ has finitely many covers with respect to the kappa order. Furthermore, we prove that every semibrick in $\modd \Lambda$ is a finite set if and only if every chain of wide subcategories of $\modd \Lambda$ is eventually constant. We also show that $\Lambda$ is brick-finite if and only if every chain of widely generated torsion classes of $\modd \Lambda$ is eventually constant. Finally, we show that $\Lambda$ is brick-finite if and only if every cofinally closed monobrick is a cofinal closure of some semibrick.