Skip to main content
SpringerLink
Log in

Abelian class groups of reductive group schemes

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce the abelian class group C ab (G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its basic properties. For example, we show that if the fraction field of A is a global field without real primes, then there exists a surjection C(G) ↠ C ab (G), where C(G) is the class set of G.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Borovoi, Abelian Galois cohomology of reductive groups, Memoirs of the American Mathematical Society 132 (1998).

  2. M. Borovoi and J. van Hamel, Extended Picard complexes and linear algebraic groups, Journal für die Reine und Angewandte Mathematik 627 (2009), 53–82.

    MATH  Google Scholar 

  3. S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  4. B. Brahm, Néron-Modelle algebraischer Tori, Schriftenreihe des Mathematischen Instituts der Universität Münster, Vol. 31, Universität Münster, Münster, 2004.

    Google Scholar 

  5. L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque 225 (1994).

  6. F. Bruhat and J. Tits, Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 34 (1987), 671–698.

    MathSciNet  MATH  Google Scholar 

  7. J.-L. Colliot-Thélène, Résolutions flasques des groupes linéaires connexes, Journal für die Reine und Angewandte Mathematik 618 (2008), 77–133.

    MATH  Google Scholar 

  8. J.-L. Colliot-Thélène and J.-J. Sansuc, R-équivalence sur les tores, Annales Scientifiques de l’École Normale Supérieure 10 (1977), 175–229.

    MATH  Google Scholar 

  9. J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori: applications, Journal of Algebra 106 (1987), 148–205.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Demazure and A. Grothendieck (eds.), Schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962–64 (SGA 3), Lecture Notes in Mathematics, Vols. 151–153, Springer, Berlin-Heidelberg-New York, 1972.

    Google Scholar 

  11. C. Demarche, Le défaut d’approximation forte dans les groupes linéaires connexes, Proceedings of the London Mathematical Society 102 (2011), 563–597.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Demarche, Théorèmes de dualité pour les complexes de tores, arXiv:0906.3453v1.

  13. J. Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Vol. 179, Springer-Verlag, Berlin-New York, 1971.

    MATH  Google Scholar 

  14. C. D. González-Avilés, Arithmetic duality theorems for 1-motives over function fields, Journal für die Reine und Angewandte Mathematik 632 (2009), 203–231.

    MATH  Google Scholar 

  15. C. D. González-Avilés, On Néron-Raynaud class groups of tori and the Capitulation Problem, Journal für die Reine und Angewandte Mathematik 648 (2010), 149–182.

    MATH  Google Scholar 

  16. C. D. González-Avilés, Quasi-abelian crossed modules and nonabelian cohomology, Journal of Algebra 369 (2012), 235–255.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. D. González-Avilés, Flasque resolutions of reductive group schemes, arXiv:1112.6020v1.

  18. A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Étude locale des schémas et des morphismes de schémas, Quatrième partie (EGA IV), Publications Mathématiques. lnsititute de Hautes Études Scientifiques 20 (1964), 52–59.

    Google Scholar 

  19. A. Grothendieck and J. Verdier (eds.), Théorie de Topos et Cohomologie Etale des Schémas, Séminaire de Géométrie Algébrique du Bois Marie 1963–64 (SGA 4III), Lecture Notes in Mathemtics, Vol. 305, Springer, Berlin-Heidelberg-New York, 1972.

  20. D. Harari and T. Szamuely, Arithmetic duality theorems for 1-motives, Journal für die Reine und Angewandte Mathematik 578 (2005), 93–128 and Errata: available from http://www.renyi.hu/~szamuely.

    MathSciNet  MATH  Google Scholar 

  21. G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen, III, Journal für die Reine und Angewandte Mathematik 274–275 (1975), 125–138.

    MathSciNet  Google Scholar 

  22. C. U. Jensen, Les Foncteurs Dérivés de \(\underleftarrow {\lim }\) et leurs Applications en Théorie des Modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Heidelberg, 1972.

    Google Scholar 

  23. S. Lang, Algebraic groups over finite fields. American Journal of Mathematics 78 (1975), 555–563.

    Article  Google Scholar 

  24. J. S. Milne, Étale Cohomology, Princeton University Press, Princeton, 1980.

    MATH  Google Scholar 

  25. J. S. Milne, Arithmetic Duality Theorems, Second Edn. (electronic version), 2006.

  26. Ye. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, in Algebraic K-theory: Connections with Geometry and Topology (Lake Louise, AB, 1987), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 279, Kluwer Academic Publ., Dordrecht, 1989, pp. 241–342.

    Chapter  Google Scholar 

  27. J. Oesterlé, Nombres de Tamagawa et groupes unipotentes en caractérisque p, Inventiones Mathematicae 78 (1984), 13–88.

    Article  MathSciNet  MATH  Google Scholar 

  28. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press Inc., New York, 1994.

    MATH  Google Scholar 

  29. J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, Journal für die Reine und Angewandte Mathematik 327 (1981), 12–80.

    MathSciNet  MATH  Google Scholar 

  30. J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, New York, 1986.

    Google Scholar 

  31. N. Q. Thǎńg, Corestriction Principle for non-abelian cohomology of reductive group schemes over Dedekind rings of integers of local and global fields, preprint (2008).

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, La Serena, Chile

    Cristian D. González-Avilés

Authors
  1. Cristian D. González-Avilés
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristian D. González-Avilés.

Additional information

The author is partially supported by Fondecyt grant 1080025.

Rights and permissions

Reprints and permissions

About this article

Cite this article

González-Avilés, C.D. Abelian class groups of reductive group schemes. Isr. J. Math. 196, 175–214 (2013). https://doi.org/10.1007/s11856-012-0147-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0147-4

Keywords

Search

Navigation