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

Gustavo Jasso

doi:10.5802/crmath.655

Research article - Representation theory

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

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
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.

Received: 2024-02-23
Revised: 2024-05-15
Accepted: 2024-06-03
Published online: 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

Author's affiliations:

Gustavo Jasso ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
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     year = {2024},
     doi = {10.5802/crmath.655},
     zbl = {07945487},
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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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