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Magnitude of module categories and special biserial algebras — arXiv:2610.00413

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.00413
License (as archived): CC0-1.0
Archived: 2026-10-06T17:41:57.240Z
SHA-256 of the archived text: dc3f22321f5f9d7cbb8c0c707dbf4bd9de5531230eddce8d99974a80ea161055
Stable URL of this page: https://metraipolis.com/documento/dc_8fa76129cfca

How to cite

Magnitude of module categories and special biserial algebras — arXiv:2610.00413 — arXiv (en), https://arxiv.org/abs/2610.00413 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_8fa76129cfca (Metraipolis archive dc_8fa76129cfca, sha256 dc3f22321f5f9d7cbb8c0c707dbf4bd9de5531230eddce8d99974a80ea161055)

BibTeX

@misc{dc8fa76129cfca,
  title = {Magnitude of module categories and special biserial algebras — arXiv:2610.00413},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.00413},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_8fa76129cfca; sha256 dc3f22321f5f9d7cbb8c0c707dbf4bd9de5531230eddce8d99974a80ea161055},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.00413. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander--Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-10-02 in math.CO).
Categorie arXiv: Representation Theory (math.RT) ; Combinatorics (math.CO); Category Theory (math.CT); Rings and Algebras (math.RA).
Pagina del lavoro: https://arxiv.org/abs/2610.00413 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T16:06:55.895Z dalla pagina https://arxiv.org/catchup/math.CO/2026-10-02?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).