Translation-invariant line bundles on linear algebraic groups
Author:
Zev Rosengarten
Journal:
J. Algebraic Geom. 30 (2021), 433-455
DOI:
https://doi.org/10.1090/jag/753
Published electronically:
March 16, 2020
MathSciNet review:
4283548
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Abstract |
References |
Additional Information
Abstract: We study the Picard groups of connected linear algebraic groups and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.
References
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References
- Raphaël Achet, The Picard group of the forms of the affine line and of the additive group, J. Pure Appl. Algebra 221 (2017), no. 11, 2838–2860. MR 3655706, DOI 10.1016/j.jpaa.2017.02.003
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164 (French). MR 181643, DOI 10.1007/BF02566948
- J.-L. Colliot-Thélène, Lectures on linear algebraic groups, 2007, accessed at http://www.math.u-psud.fr/\textasciitilde colliot/Beijing Lectures2Juin07.pdf.
- Brian Conrad, Finiteness theorems for algebraic groups over function fields, Compos. Math. 148 (2012), no. 2, 555–639. MR 2904198, DOI 10.1112/S0010437X11005665
- Brian Conrad, The structure of solvable groups over general fields, Autours des schémas en groupes. Vol. II, Panor. Synthèses, vol. 46, Soc. Math. France, Paris, 2015, pp. 159–192 (English, with English and French summaries). MR 3525596
- Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, 2nd ed., New Mathematical Monographs, vol. 26, Cambridge University Press, Cambridge, 2015. MR 3362817, DOI 10.1017/CBO9781316092439
- Schémas en groupes. I, II, III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, Vols. 151, 152, 153, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458, MR 0274459, MR 0274460
- A. Grothendieck, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1960/67) (French). MR 217083, MR 217084, MR 217085, MR 0163911, MR 0173675, MR 0199181, MR 0217086, MR 0238860
- Nguyêñ Duy Tân, On Galois cohomology of unipotent algebraic groups over local fields, J. Algebra 344 (2011), 47–59. MR 2831927, DOI 10.1016/j.jalgebra.2011.07.021
- Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, Unipotent algebraic groups, Lecture Notes in Mathematics, Vol. 414, Springer-Verlag, Berlin-New York, 1974. MR 0376696
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5 Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
- Joseph Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique $p$, Invent. Math. 78 (1984), no. 1, 13–88 (French). MR 762353, DOI 10.1007/BF01388714
- Z. Rosengarten, Tate duality in positive dimension over function fields, available at https://arxiv.org/pdf/1805.00522.pdf, 2018.
- Peter Russell, Forms of the affine line and its additive group, Pacific J. Math. 32 (1970), 527–539. MR 265367
- J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 (French). MR 631309, DOI 10.1515/crll.1981.327.12
- K. Schwede, Generalized divisors and reflexive sheaves, accessed at https://www.math.utah.edu/\textasciitilde schwede/MichiganClasses/math632/GeneralizedDivisors.pdf.
- Jean-Pierre Serre, Algebraic groups and class fields, with translated from the French, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. MR 918564, DOI 10.1007/978-1-4612-1035-1
- Burt Totaro, Pseudo-abelian varieties, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 693–721 (English, with English and French summaries). MR 3185350, DOI 10.24033/asens.2199
Additional Information
Zev Rosengarten
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, 91904, Jerusalem, Israel
Email:
zevrosengarten@gmail.com
Received by editor(s):
January 3, 2019
Received by editor(s) in revised form:
January 27, 2019, August 5, 2019, August 14, 2019, and January 31, 2020
Published electronically:
March 16, 2020
Additional Notes:
While completing this work, the author was supported by an ARCS Scholar Award, a Ric Weiland Graduate Fellowship, and a Zuckerman Postdoctoral Scholarship.
Article copyright:
© Copyright 2020
University Press, Inc.
