Comptes Rendus
Géométrie algébrique
Picard Groups of Algebraic Groups and an Affineness Criterion
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 559-564.

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.

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DOI : 10.5802/crmath.419
Classification : 14L10, 14L15, 14L17, 14L40, 20G15

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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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/

[1] Armand Borel Linear Algebraic Groups, Graduate Texts in Mathematics, 126, Springer, 1991 | Zbl

[2] Brian Conrad Units On Product Varieties, 2006 (available at http://math.stanford.edu/~conrad/papers/unitthm.pdf)

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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

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[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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