Skip to main content
SpringerLink
Log in

Lagrangianity for log extendable overconvergent F-isocrystals

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In the framework of Berthelot’s theory of arithmetic \({\mathcal {D}}\)-modules, we prove that Berthelot’s characteristic variety associated with a holonomic \({\mathcal {D}}\)-modules endowed with a Frobenius structure has pure dimension. As an application, we get the lagrangianity of the characteristic variety of a log extendable overconvergent F-isocrystal.

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. Abe, T.: Rings of microdifferential operators for arithmetic \(\fancyscript {D}\)-modules. Rend. Sem. Mat. Univ. Padova 131(1), 89–149 (2014)

    Article  Google Scholar 

  2. Berthelot, P.: \({\cal{d}}\)-modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. (4) 29(2), 185–272 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berthelot, P.: \(\cal{D}\)-modules arithmétiques. II. Descente par Frobenius. Mém. Soc. Math. Fr. (N.S.) (81), vi+136 (2000)

  4. Berthelot, P.: Introduction à la théorie arithmétique des \(\cal{D}\)-modules, Astérisque (2002), 279, 1–80, Cohomologies \(p\)-adiques et applications arithmétiques, II

  5. Caro, D.: Dévissages des \(F\)-complexes de \(\cal{D}\)-modules arithmétiques en \(F\)-isocristaux surconvergents. Invent. Math. 166(2), 397–456 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caro, D.: Fonctions L associées aux \(\cal{D}\)-modules arithmétiques. Cas des courbes Compos. Math. 142(01), 169–206 (2006)

    Article  MATH  Google Scholar 

  7. Caro, D.: Overconvergent log-isocrystals and holonomy. (Log-isocristaux surconvergents et holonomie.). Compos. Math. 145(6), 1465–1503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caro, D.: Betti number estimates in \(p\)-adic cohomology, ArXiv Mathematics e-prints (2015)

  9. Caro, D., Tsuzuki, N.: Overholonomicity of overconvergent \(F\)-isocrystals over smooth varieties. Ann. Math. 176(2), 747–813 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 (1961)

    Google Scholar 

  11. Hotta, R., Takeuchi, K. Tanisaki, T.: \(D\)-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi

  12. Kashiwara, M.: \(B\)-functions and holonomic systems. Rationality of roots of \(B\)-functions. Invent. Math. 38(1): 33–53 (1976/77)

  13. Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals, IV: local semistable reduction at nonmonomial valuations. Compos. Math. 147(2), 467–523 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Laumon, G.: Sur la catégorie dérivée des \({\cal{D}}\)-Modules filtrés, Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016. Springer, Berlin (1983)

    Google Scholar 

  15. Laumon, G.: Transformations canoniques et spécialisation pour les \({\cal{D}}\)-modules filtrés, Astérisque (130):56–129, differential systems and singularities (Luminy, 1983) (1985)

  16. Malgrange, B.: Caractérisation homologique de la dimension, Séminaire Opérateurs différentiels et pseudo-différentiels, Grenoble (1975-1976)

  17. Matsumura, H.: Commutative Algebra, second éd. Mathematics Lecture Note Series, vol. 56. Benjamin/Cummings Publishing Co., Inc., Reading (1980)

    MATH  Google Scholar 

  18. Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Société Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960–61. [Geometric Algebra Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin]

  19. Virrion, A.: Dualité locale et holonomie pour les \(\cal{D}\)-modules arithmétiques. Bull. Soc. Math. France 128(1), 1–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Weibel, Charles A.: An Introduction to Homological Algebra, Volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, Campus 2, 14032, Caen Cedex, France

    Daniel Caro

Authors
  1. Daniel Caro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Caro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caro, D. Lagrangianity for log extendable overconvergent F-isocrystals. Math. Z. 287, 325–339 (2017). https://doi.org/10.1007/s00209-016-1827-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1827-2

Keywords

Search

Navigation