A characterisation of higher torsion classes
Let $\mathcal {A}$ be an abelian length category containing a d-cluster tilting subcategory $\mathcal {M}$ . We prove that a subcategory of $\mathcal {M}$ is a d-torsion class if and only if it is closed under d-extensions and d-quotients. This generalises an important result for...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000732/type/journal_article |
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| author | Jenny August Johanne Haugland Karin M. Jacobsen Sondre Kvamme Yann Palu Hipolito Treffinger |
| author_facet | Jenny August Johanne Haugland Karin M. Jacobsen Sondre Kvamme Yann Palu Hipolito Treffinger |
| author_sort | Jenny August |
| collection | DOAJ |
| description | Let
$\mathcal {A}$
be an abelian length category containing a d-cluster tilting subcategory
$\mathcal {M}$
. We prove that a subcategory of
$\mathcal {M}$
is a d-torsion class if and only if it is closed under d-extensions and d-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the d-torsion classes in
$\mathcal {M}$
form a complete lattice. Moreover, we use the characterisation to classify the d-torsion classes associated to higher Auslander algebras of type
$\mathbb {A}$
, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras. |
| format | Article |
| id | doaj-art-d6f4418aca6146b4b41c9c31bd5364d0 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-d6f4418aca6146b4b41c9c31bd5364d02025-02-07T07:50:24ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.73A characterisation of higher torsion classesJenny August0https://orcid.org/0000-0001-7999-3858Johanne Haugland1https://orcid.org/0000-0003-3128-4715Karin M. Jacobsen2https://orcid.org/0000-0001-9535-6613Sondre Kvamme3https://orcid.org/0000-0002-4056-2417Yann Palu4https://orcid.org/0009-0008-1761-593XHipolito Treffinger5https://orcid.org/0000-0002-4573-9496Department of Mathematics, Aarhus Universitet, Ny Munkegade 118, DK-8000 Aarhus C, Denmark; E-mail:Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway; E-mail:Department of Mathematics, Aarhus Universitet, Ny Munkegade 118, DK-8000 Aarhus C, Denmark; E-mail:Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, NorwayLAMFA, Université de Picardie Jules Verne, 33, rue Saint-Leu 80039 Amiens, France; E-mail:IMAS-CONICET, Pabellon I – Ciudad Universitaria, Buenos Aires, 1428, Argentina; E-mail:Let $\mathcal {A}$ be an abelian length category containing a d-cluster tilting subcategory $\mathcal {M}$ . We prove that a subcategory of $\mathcal {M}$ is a d-torsion class if and only if it is closed under d-extensions and d-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the d-torsion classes in $\mathcal {M}$ form a complete lattice. Moreover, we use the characterisation to classify the d-torsion classes associated to higher Auslander algebras of type $\mathbb {A}$ , and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.https://www.cambridge.org/core/product/identifier/S2050509424000732/type/journal_article16S9018E4018E1018G2518G99 |
| spellingShingle | Jenny August Johanne Haugland Karin M. Jacobsen Sondre Kvamme Yann Palu Hipolito Treffinger A characterisation of higher torsion classes Forum of Mathematics, Sigma 16S90 18E40 18E10 18G25 18G99 |
| title | A characterisation of higher torsion classes |
| title_full | A characterisation of higher torsion classes |
| title_fullStr | A characterisation of higher torsion classes |
| title_full_unstemmed | A characterisation of higher torsion classes |
| title_short | A characterisation of higher torsion classes |
| title_sort | characterisation of higher torsion classes |
| topic | 16S90 18E40 18E10 18G25 18G99 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000732/type/journal_article |
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