Finiteness of semibricks and brick-finite algebras

TL;DR

This paper establishes a bijection between certain torsion classes and semibricks in module categories of finite-dimensional algebras, providing new characterizations of brick-finite algebras and proving Enomoto's conjecture.

Contribution

It introduces explicit bijections and equivalent conditions for brick-finiteness, and proves a conjecture relating wide subcategories and finiteness properties in module categories.

Findings

Brick-finite algebras are characterized by chains of wide subcategories becoming constant.

Every torsion class in a brick-finite algebra has finitely many covers.

All semibricks in a brick-finite algebra are finite sets.

Abstract

For a finite-dimensional algebra {\Lambda}, we establish an explicit bijection between widely generated torsion(-free) classes and semibricks in mod {\Lambda}. 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 {\Lambda} is brick-finite if and only if any chain of wide subcategories of mod {\Lambda} becomes eventually constant, if and only if any torsion class in mod {\Lambda} has finitely many covers, if and only if every semibrick in mod {\Lambda} is a finite set. Thus, we give a proof of Enomoto's conjecture (Adv. Math., 393 (2021), 108113). As a consequence, we show that {\Lambda} is brick-finite if and only if every wide subcategory closed under coproducts of Mod {\Lambda} is closed under products, if and only if every wide subcategory of mod {\Lambda} is functorially finite. This gives a positive answer to the question posed by Angeleri H\"ugel and Sentieri (J. Algebra, 664 (2025), 164-205).