Picard Groups of Algebraic Groups and an Affineness Criterion

Rosengarten, Zev

doi:10.5802/crmath.419

Géométrie algébrique

Picard Groups of Algebraic Groups and an Affineness Criterion

¹Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, 91904, Jerusalem, Israel

Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 559-564

Abstract (VO)
Résumé

We prove that an algebraic group over a field $k$ is affine precisely when its Picard group is torsion, and show that in this case the Picard group is finite when $k$ is perfect, and the product of a finite group of order prime to $p$ and a $p$ -primary group of finite exponent when $k$ is imperfect of characteristic $p$ .

Nous prouvons qu’un groupe algébrique sur un corps $k$ est affine si et seulement si son groupe de Picard est de torsion, et que dans ce cas, le groupe de Picard est fini si $k$ est parfait, et le produit d’un groupe fini d’ordre premier à $p$ par un $p$ -groupe d’exposant fini lorsque $k$ est imparfait de caractéristique $p$ .

Reçu le : 2022-05-10
Révisé le : 2022-09-06
Accepté le : 2022-09-07
Publié le : 2023-02-01

DOI : 10.5802/crmath.419

Classification : 14L10, 14L15, 14L17, 14L40, 20G15

Affiliations des auteurs :

Rosengarten, Zev ¹

¹ Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, 91904, Jerusalem, Israel

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Picard {Groups} of {Algebraic} {Groups} and an {Affineness} {Criterion}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {559--564},
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Rosengarten, Zev. Picard Groups of Algebraic Groups and an Affineness Criterion. Comptes Rendus. Mathématique, Tome 361 (2023) no. G2, pp. 559-564. doi: 10.5802/crmath.419

Bibliographie
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