Skip to main content
SpringerLink
Log in

On Kashiwara’s equivalence in positive characteristic

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

Bezrukavnikov, Mirkovic and Rumynin have recently obtained a derived version of the Beilinson-Bernstein localization theorem for the the Lie algebra of a semisimple algebraic group in positive characteristic p using the sheaf of rings of crystalline differential operators. A central reduction of is the first term of the p-filtration of the ring of the standard differential operators. We observe that the direct image functors of -modules on smooth varieties do not behave well; Kashiwara’s equivalence for a closed immersion fails, for example. On the other hand, we find that the direct image as -modules of the structure sheaf of the Frobenius neighbourhood of a point in each Chevalley-Bruhat cell under its inclusion in the flag variety realizes upon taking global sections an infinitesimal Verma module.

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. Beilinson, A., Bernstein, J.: A proof of Jantzen conjectures. Advances in Soviet Math. 16 (part 1), 1–49 (1993)

    MATH  Google Scholar 

  2. Berthelot, P.: Cohomologie Cristalline des Schémas de Caracteristique p>0. Lecture Notes in Math. 407, Berline/Heidelberg 1974 (Springer)

  3. Berthelot, P.: -modules arithmétiques I. Opérateurs différentiels de niveau fini. Ann. scient. Éc. Norm. Sup. 29, 185–272 (1996)

    Google Scholar 

  4. Berthelot, P.: -modules arithmétiques II. Descente par Frobenius. Mém. de la Soc. Math. France 81, (2000)

  5. Bezrukavnikov, R., Mirkovic, I., Rumynin, D.: Localization of modules for a semisimple Lie algebra in prime characteristic. To appear

  6. Grothendieck, A., Dieudonné, J.: Éléments de Géométrie Algébrique IV, Pub. Math. no. 32, IHES 1967

  7. Haastert, B.: On direct and inverse images of -modules in prime characteristic. Manusc. Math. 62, 341–354 (1988)

    Google Scholar 

  8. Jantzen, J. C.: Representations of Algebraic Groups. Pure and Applied Mathematics 131, Academic Press, Boston etc. 1987

  9. Kashiwara M.: Algebraic study of systems of partial differential equations. Master’s Thesis, Tokyo Univ., 1970

Download references

Author information

Authors and Affiliations

  1. Osaka City University, Department of Mathematics, 558-8585, Osaka, Japan

    Masaharu Kaneda

Authors
  1. Masaharu Kaneda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masaharu Kaneda.

Additional information

Revised version: 27 April 2004

Supported in part by JSPS Grant in Aid for Scientific Research

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaneda, M. On Kashiwara’s equivalence in positive characteristic. manuscripta math. 114, 457–468 (2004). https://doi.org/10.1007/s00229-004-0473-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-004-0473-x

Keywords

Search

Navigation