Finiteness of semibricks and brick-finite algebras
Alireza Nasr-Isfahani
TL;DR
The paper proves that a finite-dimensional algebra $Λ$ is brick-finite if and only if several equivalent conditions hold, notably that every chain of wide subcategories stabilizes, every torsion class has finitely many covers, and every semibrick is finite. It achieves this through a detailed lattice-theoretic framework based on the $κ$-order on the torsion lattice and canonical join representations, and by establishing a bijection between widely generated torsion($-$free) classes and semibricks via minimal extending/co-extending modules. This yields a complete proof of Enomoto's conjecture and leads to further equivalences showing that wide subcategories with coproducts closed are closed under products, and that all wide subcategories of mod $Λ$ are functorially finite, thereby answering questions of Angeleri Hügel–Sentieri. The results also connect semibricks to cofinally closed monobricks and provide injections (and in brick-finite cases bijections) between these objects, enriching the interplay between torsion theory, representation theory, and lattice theory.
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).