Comptes Rendus
Article de recherche - Géométrie algébrique
A note on Azumaya algebras and one-forms
[Note sur les algèbres d’Azumaya et les $1$-formes]
Siqing Zhang1 orcid
1Department of Mathematics, Yale University, New Haven, CT, 06511, USA
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 545-548

The crystalline differential operators on a smooth variety $X$ give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist $X^{\prime }$. In some cases, this Azumaya algebra splits when restricted to finite covers of $X^{\prime }$. In this short note, we show that, whenever $X$ has a non-closed global one-form, there is a degree-$1$ cover of $X^{\prime }$ on which the Azumaya algebra does not split, answering a question of Sasha Petrov.

Les opérateurs différentiels cristallins sur une variété lisse $X$ donnent naissance à une algèbre d’Azumaya non décomposable sur le fibré cotangent de la torsion de Frobenius $X^{\prime }$. Dans certains cas, cette algèbre d’Azumaya se décompose lorsqu’elle est restreinte aux recouvrements finis de $X^{\prime }$. Dans cette brève note, nous montrons que, chaque fois que $X$ possède une $1$-forme globale non fermée, il existe un recouvrement de degré $1$ de $X^{\prime }$ sur lequel l’algèbre d’Azumaya ne se décompose pas, répondant ainsi à une question posée par Sasha Petrov.

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DOI : 10.5802/crmath.844
Classification : 14G17
Keywords: Azumaya algebra, positive characteristics
Mots-clés : Algèbre d’Azumaya, caractéristique positive

Siqing Zhang  1

1 Department of Mathematics, Yale University, New Haven, CT, 06511, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Siqing Zhang. A note on Azumaya algebras and one-forms. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 545-548. doi: 10.5802/crmath.844
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     language = {en},
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