Selecta Mathematica

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

On Beilinson’s equivalence for p-adic cohomology

  • Tomoyuki Abe
  • Daniel Caro


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 


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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.


  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).
  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.
  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

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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

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