Finiteness of semibricks and brick-finite algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912905624027136 |
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| author | Nasr-Isfahani, Alireza |
| author_facet | Nasr-Isfahani, Alireza |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness of semibricks and brick-finite algebras Nasr-Isfahani, Alireza Representation Theory 16G10, 16S90, 18E10, 05E10, 06A07 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). |
| title | Finiteness of semibricks and brick-finite algebras |
| topic | Representation Theory 16G10, 16S90, 18E10, 05E10, 06A07 |
| url | https://arxiv.org/abs/2511.12198 |