Finiteness of semibricks and brick-finite algebras

Abstract: For a finite-dimensional algebra Λ, we establish an explicit bijection between widely generated torsion(-free) classes and semibricks in mod Λ. Using the kappa order on the lattice of torsion classes with canonical join representations, we provide several equivalent conditions for brick-finite algebras. We show that Λ is brick-finite if and only if any chain of wide subcategories of mod Λ becomes eventually constant, if and only if any torsion class in mod Λ has finitely many covers, if and only if every semibrick in mod Λ is a finite set. Thus, we give a proof of Enomoto's conjecture (Adv. Math., 393 (2021), 108113). As a consequence, we show that Λ is brick-finite if and only if every wide subcategory closed under coproducts of Mod Λ is closed under products, if and only if every wide subcategory of mod Λ is functorially finite. This gives a positive answer to the question posed by Angeleri Hügel and Sentieri (J. Algebra, 664 (2025), 164-205).