On a theorem of B. Keller on Yoneda algebras of simple modules

Gustavo Jasso

doi:10.5802/crmath.655

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On a theorem of B. Keller on Yoneda algebras of simple modules
[Sur un théorème de B. Keller sur les algèbres de Yoneda de modules simples]

Gustavo Jasso ¹

¹ Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1449-1453.

Résumé
Abstract

Un théorème de Keller stipule que l’algèbre de Yoneda des modules simples sur une algèbre de dimension finie est générée en degrés cohomologiques $0$ et $1$ comme une $A_{\infty}$ -algèbre minimale. Nous prouvons une extension du théorème de Keller aux catégories de longueur abélienne en réduisant le problème à une classe particulière d’algèbres de Nakayama, où l’affirmation peut être démontrée par un calcul direct.

A theorem of Keller states that the Yoneda algebra of the simple modules over a finite-dimensional algebra is generated in cohomological degrees $0$ and $1$ as a minimal $A_{\infty}$ -algebra. We provide a proof of an extension of Keller’s theorem to abelian length categories by reducing the problem to a particular class of Nakayama algebras, where the claim can be shown by direct computation.

Reçu le : 2024-02-23
Révisé le : 2024-05-15
Accepté le : 2024-06-03
Publié le : 2024-11-14

DOI : 10.5802/crmath.655

Classification : 18G70, 16G20, 16G70
Keywords: Yoneda algebras, simple modules, Nakayama algebras, $A_\infty $-algebras
Mot clés : Algèbres de Yoneda, modules simples, algèbres de Nakayama, $A_\infty $-algèbres

Affiliations des auteurs :

Gustavo Jasso ¹

¹ Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On a theorem of {B.~Keller} on {Yoneda} algebras of simple modules},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1449--1453},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.655},
     language = {en},
}

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Gustavo Jasso. On a theorem of B. Keller on Yoneda algebras of simple modules. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1449-1453. doi : 10.5802/crmath.655. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.655/

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