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.

Abstract (VO)
Résumé

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.

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.

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

Zbl

DOI : 10.5802/crmath.655

Classification : 18G70, 16G20, 16G70
Keywords: Yoneda algebras, simple modules, Nakayama algebras,

A_{\infty}

-algebras
Mots-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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     author = {Gustavo Jasso},
     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},
     zbl = {07945487},
     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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