Advertisement

Selecta Mathematica

, Volume 24, Issue 2, pp 591–608 | Cite as

On Beilinson’s equivalence for p-adic cohomology

  • Tomoyuki Abe
  • Daniel Caro
Article

Abstract

In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic \({\mathcal {D}}\)-modules and the derived category of arithmetic \({\mathcal {D}}\)-modules whose cohomologies are holonomic.

Mathematics Subject Classification

14F10 14F30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The first author (T.A.) was supported by Grant-in-Aid for Young Scientists (B) 25800004. The second author (D.C.) thanks Antoine Chambert-Loir for his suggestion to consider the comparison of Euler characteristics in the p-adic context. The second author (D.C) was supported by the I.U.F.

References

  1. 1.
    Abe, T.: Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \({\cal{D}}\)-modules. Rend. Semin. Mat. Univ. Padova 131, 89–149 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abe, T.: Langlands correspondence for isocrystals and existence of crystalline companion for curves, preprint, arXiv:1310.0528
  3. 3.
    Abe, T., Caro, D.: Theory of weights in \(p\)-adic cohomology, preprint, arXiv:1303.0662
  4. 4.
    Abe, T., Marmora, A.: On \(p\)-adic product formula for epsilon factors. JIMJ 14(2), 275–377 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beilinson, A.: On the derived category of perverse sheaves. Lecture Notes in Mathematics, vol. 1289, pp. 27–41. Springer, Berlin (1987)Google Scholar
  6. 6.
    Beilinson, A.: How to glue perverse sheaves. Lecture Notes in Mathematics, vol. 1289, pp. 42–51. Springer, Berlin (1987)Google Scholar
  7. 7.
    Beilinson, A., Bernstein, J.: A proof of Jantzen conjectures. Adv. Soviet Math. 16, 1–50 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres. Première partie (version provisoire 1991), Prépublication IRMR 96-03 (1996)Google Scholar
  9. 9.
    Berthelot, P.: \({\cal{D}}\)-modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. 29, 185–272 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Berthelot, P.: \({\cal{D}} \)-modules arithmétiques. II. Descente par Frobenius. Mém. Soc. Math. Fr. (N.S.), p. vi+136 (2000)Google Scholar
  11. 11.
    Berthelot, P.: Introduction à la théorie arithmétique des \(\cal{D}\)-modules, Astérisque, no. 279, pp. 1–80. Cohomologies \(p\)-adiques et applications arithmétiques, II (2002)Google Scholar
  12. 12.
    Caro, D.: Dévissages des \(F\)-complexes de \(\cal{D}\)-modules arithmétiques en \(F\)-isocristaux surconvergents. Invent. Math. 166, 397–456 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caro, D.: \({\cal{D}}\)-modules arithmétiques surholonomes. Ann. Sci. École Norm. Sup. 42, 141–192 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Caro, D.: Holonomie sans structure de Frobenius et critères d’holonomie. Ann. Inst. Fourier 61, 1437–1454 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Caro, D.: Stabilité de l’holonomie sur les variétés quasi-projectives. Compos. Math. 147, 1772–1792 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Caro, D.: Sur la préservation de la surconvergence par l’image directe d’un morphisme propre et lisse. Ann. Sci. Éc. Norm. Supér. (4) 48(1), 131–169 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Caro, D.: Sur la stabilité par produit tensoriel de complexes de \({\cal{D}}\)-modules arithmétiques. Manuscr. Math. 147, 1–41 (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Caro, D.: La surcohérence entraîne l’holonomie. Bull. Soc. Math. Fr. 144(3), 429–475 (2016)CrossRefzbMATHGoogle Scholar
  19. 19.
    Caro, D.: Systèmes inductifs cohérents de \({\cal{D}}\)-modules arithmétiques logarithmiques, stabilité par opérations cohomologiques. Doc. Math. 21, 1515–1606 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Caro, D.: Unipotent monodromy and arithmetic \({\cal{D}}\)-modules. Manuscr. Math. (2017).  https://doi.org/10.1007/s00229-017-0959-y
  21. 21.
    Christol, G., Mebkhout, Z.: Sur le théorème de l’indice des équations différentielles \(p\)-adiques IV. Invent. Math. 143, 629–672 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Crew, R.: Arithmetic \({\cal{D}}\)-modules on the unit disk. Compos. Math. 48, 227–268 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals. I. Unipotence and logarithmic extensions. Compos. Math. 143, 1164–1212 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kedlaya, K.S.: Semistable reduction for overconvergent \(F\)-isocrystals, IV: local semistable reduction at nonmonomial valuations. Compos. Math. 147, 467–523 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Laumon, G.: Comparaison de caractéristiques d’Euler–Poincaré en cohomologie \(l\)-adique. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 209–212 (1981)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lichtenstein, S.: Vanishing cycles for algebraic \({\cal{D}}\)-modules, thesis. http://www.math.harvard.edu/~gaitsgde/grad_2009/Lichtenstein(2009).pdf
  27. 27.
    Virrion, A.: Dualité locale et holonomie pour les \(\cal{D}\)-modules arithmétiques. Bull. Soc. Math. Fr. 128(1), 1–68 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Kavli Institute for the Physics and Mathematics of the Universe (WPI)The University of TokyoKashiwaJapan
  2. 2.Laboratoire de Mathématiques Nicolas Oresme (LMNO)Université de Caen, Campus 2Caen CedexFrance

Personalised recommendations