Skip to main content
SpringerLink
Log in

A Short Exact Sequence

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Let R be a semilocal integral Dedekind domain and K be its fraction field. Let μ : G → T be an R-group scheme morphism between reductive R-group schemes that is smooth as a scheme morphism. Assume that T is an R-torus. Then the map T(R)/μ(G(R)) → T(K)/μ(G(K)) is injective, and a certain purity theorem is true. These and other results are derived from an extended form of the Grothendieck–Serre conjecture proven in the present paper for rings R as above. Bibliography: 21 titles.

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. V. Chernousov, P. Gille, and A. Pianzola, “A classification of torsors over Laurent polynomial rings,” Comment. Math. Helv., 92, 37–55 (2017).

    Article  MathSciNet  Google Scholar 

  2. J.-L. Colliot-Thélène and M. Ojanguren, “Éspaces principaux homogènes localement triviaux,” Publ. Math. Inst. Hautes Études Sci., 75, No. 2, 97–122 (1992).

    Article  Google Scholar 

  3. J.-L. Colliot-Thélène and J.-J. Sansuc, “Principal homogeneous spaces under flasque tori: Applications,” J. Algebra, 106, 148–205 (1987).

    Article  MathSciNet  Google Scholar 

  4. M. Demazure and A. Grothendieck (eds.), Schémas en Groupes, Lect. Notes Math., 151–153, Springer-Verlag, Berlin–Heidelberg–New York (1970).

    Google Scholar 

  5. M. Demazure and A. Grothendieck (eds.), Structure des Sch´emas en Groupes Réductifs, Lect. Notes Math., 153, Springer-Verlag, Berlin–Heidelberg–New York (1970).

    Google Scholar 

  6. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer- Verlag, New York (1995).

    MATH  Google Scholar 

  7. R. Fedorov and I. Panin, “A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field,” Publ. Math. Inst. Hautes Études Sci., 122, 169–193 (2015).

    Article  MathSciNet  Google Scholar 

  8. A. Grothendieck, “Torsion homologique et section rationnelles,” in: Anneaux de Chow et Applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris (1958).

    Google Scholar 

  9. A. Grothendieck, “Technique de descente et theoremes d’existence en geometrie algebrique: I. Generalites. Descente par morphismes fidèlement plats,” in: Seminaire Bourbaki, Vol. 5, Exp. No. 190, Soc. Math. France, Paris (1995), pp. 299–327.

    Google Scholar 

  10. A. Grothendieck, “Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. É tude cohomologique des faisceaux cohérents. Première partie,” Publ. Math. Inst. Hautes Études Sci., 11, 5–167 (1961).

    Article  Google Scholar 

  11. A. Grothendieck, “´Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas. Seconde partie,” Publ. Math. Inst. Hautes Études Sci., 24, 5–231 (1965).

    Article  Google Scholar 

  12. Y. Nisnevich, “Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings,” C. R. Acad. Sci. Paris Sér. I, 299, No. 1, 5–8 (1984).

    MathSciNet  MATH  Google Scholar 

  13. N. Guo, “The Grotendieck–Serre conjecture over semi-local Dedekind rings,” arXiv:1902.02315v2.

  14. I. Panin, “On Grothendieck–Serres conjecture concerning principal G-bundles over reductive group schemes: II,” Izv. Math., 80, No. 4, 131–162 (2016).

    Article  MathSciNet  Google Scholar 

  15. I. Panin, “On Grothendieck–Serre conjecture concerning principal bundles,” in: Proceedings of the International Congress of Mathematicians, Vol. 1, Rio de Janeiro (2018), pp. 201–222.

  16. I. Panin, “Nice triples and Grothendieck–Serre’s conjecture concerning principal G-bundles over reductive group schemes,” Duke Math. J., 168, No. 2 (2019).

  17. I. Panin, “Two purity theorems and Grothendieck–Serre’s conjecture concerning principal G-bundles over regular semi-local rings,” arXiv:1707.01763.

  18. I. Panin, “Proof of Grothendieck–Serre conjecture on principal G-bundles over semi-local regular domains containing a finite field,” arXiv:1707.01767.

  19. I. A. Panin and A. K. Stavrova, “On the Grothendieck–Serre conjecture concerning principal G-bundles over semi-local Dedekind domains,” Zap. Nauchn. Semin. POMI, 443, 133–146 (2016).

    Google Scholar 

  20. J.-P. Serre, “Espaces fibrés algébriques,” in: Anneaux de Chow et Applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris (1958).

    Google Scholar 

  21. A. Suslin and V. Voevodsky, “Singular homology of abstract algebraic varieties,” Invent. Math., 123, No. 1, 61–94 (1996).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg Department of Steklov Institute of Mathematics, St.Petersburg, Russia

    I. Panin

Authors
  1. I. Panin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Panin.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 485, 2019, pp. 176–186.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panin, I. A Short Exact Sequence. J Math Sci 251, 419–426 (2020). https://doi.org/10.1007/s10958-020-05101-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-05101-8

Search

Navigation