Picard Groups of Algebraic Groups and an Affineness Criterion

Zev Rosengarten

doi:10.5802/crmath.419

Géométrie algébrique

Picard Groups of Algebraic Groups and an Affineness Criterion

Zev Rosengarten ¹

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

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 559-564.

Résumé
Abstract

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 $p$ .

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 $p$ .

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 :

Zev Rosengarten 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},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2023},
     doi = {10.5802/crmath.419},
     language = {en},
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Zev Rosengarten. Picard Groups of Algebraic Groups and an Affineness Criterion. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 559-564. doi : 10.5802/crmath.419. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.419/

Bibliographie
Cité par

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[4] Alexander Grothendieck Éléments de géométrie algébrique. IV : Étude locale des schémas et des morphismes de schémas, Seconde partie, Publ. Math., Inst. Hautes Étud. Sci., Volume 24 (1965), pp. 5-231 | Zbl

[5] Robert Guralnick; David B. Jaffe; Wayne Raskind; Roger Wiegand On the Picard Group: Torsion and the Kernel Induced by a Faithfully Flat Map, J. Algebra, Volume 183 (1996) no. 2, pp. 420-455 | DOI | Zbl

[6] David B. Mumford Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay byOxford University Press, 1970

[7] Zev Rosengarten Translation-Invariant Line Bundles On Linear Algebraic Groups, J. Algebr. Geom. (2020) | DOI | Zbl

[8] Zev Rosengarten Tate Duality In Positive Dimension Over Function Fields (2021) (https://arxiv.org/abs/1805.00522, to appear in Memoirs of the American Mathematical Society)

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