On the monoidal invariance of the cohomological dimension of Hopf algebras

Julien Bichon

doi:10.5802/crmath.329

Algèbre

On the monoidal invariance of the cohomological dimension of Hopf algebras

Julien Bichon ¹

¹ Université Clermont Auvergne, CNRS, LMBP, 63000 Clermont-Ferrand, France

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 561-582.

Résumé
Abstract

Nous étudions la question de l’invariance monoïdale de la dimension globale des algèbres de Hopf : si deux algèbres de Hopf ont des catégories de comodules monoïdalement équivalentes, ont-elles même dimension globale ? Nous apportons plusieurs nouvelles réponses positives dans les cas d’algèbres de Hopf lisses, cosemisimples ou de dimension finie. Nous comparons également la dimension globale et la dimension cohomologique de Gerstenhaber–Schack dans le cas cosemisimple. Un outil important pour obtenir ces divers résultats est la nouvelle notion de foncteur séparable twisté.

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber–Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

Reçu le : 2021-08-25
Révisé le : 2022-01-06
Accepté le : 2022-01-14
Publié le : 2022-05-23

DOI : 10.5802/crmath.329

Classification : 16T05, 16E40, 16E10

Affiliations des auteurs :

Julien Bichon ¹

¹ Université Clermont Auvergne, CNRS, LMBP, 63000 Clermont-Ferrand, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Julien Bichon},
     title = {On the monoidal invariance of the cohomological dimension of {Hopf} algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {561--582},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.329},
     language = {en},
}

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Julien Bichon. On the monoidal invariance of the cohomological dimension of Hopf algebras. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 561-582. doi : 10.5802/crmath.329. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.329/

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