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Peer-reviewedCC BYJournal of Pure and Applied Algebra

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

  • Xin Fang1,2
  • Mikhail Gorsky3,4
  • Yann Palu5,6
  • Pierre‐Guy Plamondon7,8
  • Matthew Pressland9,10
10 affiliations
  1. 1Lehrstuhl für Algebra und Darstellungstheorie, RWTH Aachen University, Pontdriesch 10–16, 52062 Aachen, Germany
  2. 2RWTH Aachen - RWTH Aachen University = Rheinisch-Westfälische Technische Hochschule Aachen (RWTH Aachen – Templergraben 55 – 52062 Aachen (Hausanschrift) – 52056 Aachen (Postanschrift) – Deutschland - Germany)
  3. 3Department of Mathematics, University of Vienna, Oscar-Morgenstern Platz 1, 1090 Vienna, Austria
  4. 4Department of Mathematics, University of Vienna (Department of Mathematics, University Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria - Austria)
  5. 5LAMFA, Université Picardie Jules Verne, 33 Rue Saint-Leu, 80039 Amiens, France
  6. 6LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 UPJV (Université de Picardie Jules Verne - 33 Rue Saint-Leu - 80039 AMIENS CEDEX 1 - France)
  7. 7Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États Unis, 78000 Versailles, France
  8. 8LMV - Laboratoire de Mathématiques de Versailles (Bâtiment Fermat - UFR de sciences 45 avenue des Etats-Unis 78035 VERSAILLES - France)
  9. 9School of Mathematics & Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ, United Kingdom
  10. 10School of Mathematics & Statistics, University of Glasgow (United Kingdom)
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Abstract

We extend results of Brüstle–Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

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Cite

BibTeXRIS
BibTeX
@article{fang2026extriangulated,
  title = {{Extriangulated ideal quotients, with applications to cluster theory and gentle algebras}},
  author = {Xin Fang and Mikhail Gorsky and Yann Palu and Pierre‐Guy Plamondon and Matthew Pressland},
  journal = {Journal of Pure and Applied Algebra},
  year = {2026},
  month = aug,
  doi = {10.1016/j.jpaa.2026.108370},
  url = {https://doi.org/10.1016/j.jpaa.2026.108370},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Extriangulated ideal quotients, with applications to cluster theory and gentle algebras
AU  - Xin Fang
AU  - Mikhail Gorsky
AU  - Yann Palu
AU  - Pierre‐Guy Plamondon
AU  - Matthew Pressland
PY  - 2026
DA  - 2026/08/06
JO  - Journal of Pure and Applied Algebra
DO  - 10.1016/j.jpaa.2026.108370
UR  - https://doi.org/10.1016/j.jpaa.2026.108370
AB  - We extend results of Brüstle–Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.
SN  - 0022-4049
SN  - 1873-1376
ER  -