Semibricks and Brick-finite algebras

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.