We prove that an algebraic group over a field is affine precisely when its Picard group is torsion, and show that in this case the Picard group is finite when is perfect, and the product of a finite group of order prime to and a -primary group of finite exponent when is imperfect of characteristic .
Nous prouvons qu’un groupe algébrique sur un corps 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 est parfait, et le produit d’un groupe fini d’ordre premier à par un -groupe d’exposant fini lorsque est imparfait de caractéristique .
Revised:
Accepted:
Published online:
Zev Rosengarten 1
@article{CRMATH_2023__361_G2_559_0, author = {Zev Rosengarten}, title = {Picard {Groups} of {Algebraic} {Groups} and an {Affineness} {Criterion}}, journal = {Comptes Rendus. Math\'ematique}, pages = {559--564}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.419}, language = {en}, }
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] Linear Algebraic Groups, Graduate Texts in Mathematics, 126, Springer, 1991 | Zbl
[2] Units On Product Varieties, 2006 (available at http://math.stanford.edu/~conrad/papers/unitthm.pdf)
[3] Schémas en Groupes, Lecture Notes in Mathematics, I, II, III,, Springer, 1970
[4] 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] 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] 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] Translation-Invariant Line Bundles On Linear Algebraic Groups, J. Algebr. Geom. (2020) | DOI | Zbl
[8] Tate Duality In Positive Dimension Over Function Fields (2021) (https://arxiv.org/abs/1805.00522, to appear in Memoirs of the American Mathematical Society)
Cited by Sources:
Comments - Policy