Nonabelian Hodge theory in characteristic $p$

Ogus, A.; Vologodsky, V.

doi:10.1007/s10240-007-0010-z

Nonabelian Hodge theory in characteristic

p

Ogus, A. ; Vologodsky, V.

Publications Mathématiques de l'IHÉS, Tome 106 (2007), pp. 1-138.

Résumé

Given a scheme in characteristic $p$ together with a lifting modulo $p^{2}$ , we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients.

DOI : 10.1007/s10240-007-0010-z

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     author = {Ogus, A. and Vologodsky, V.},
     title = {Nonabelian {Hodge} theory in characteristic $p$},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--138},
     publisher = {Springer},
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     year = {2007},
     doi = {10.1007/s10240-007-0010-z},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-007-0010-z/}
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Ogus, A.; Vologodsky, V. Nonabelian Hodge theory in characteristic $p$. Publications Mathématiques de l'IHÉS, Tome 106 (2007), pp. 1-138. doi : 10.1007/s10240-007-0010-z. http://archive.numdam.org/articles/10.1007/s10240-007-0010-z/

Bibliographie
Cité par

1. A. Beilinson, On the derived category of perverse sheaves, in K-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Lect. Notes Math., vol. 1289, Springer, Berlin Heidelberg New York, 1987. | MR | Zbl

2. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque, 100 (1982), 5-171 | MR | Zbl

3. P. Berthelot, A. Ogus, Notes on Crystalline Cohomology, Princeton University Press, Princeton, N.J. (1978) | MR | Zbl

4. R. Bezrukavnikov, I. Mirković, and D. Rumynin, Localization of modules for a semisimple lie algebra in prime characteristic, Ann. Math., to appear, arXiv:math RT/0205144v5.

5. A. Braverman, R. Bezrukavnikov, Geometric Langlands correspondence for $𝒟$ -modules in prime characteristic: the Gl(n) case, Pure Appl. Math. Q., 3 (2007), 153-179 | MR

6. P. Deligne, Equations Différentielles à Points Singuliers Réguliers, Springer, Berlin Heidelberg New York (1970) | MR | Zbl

7. P. Deligne, Théorie de Hodge II, Publ. Math., Inst. Hautes Étud. Sci., 40 (1972), 5-57 | Numdam | MR | Zbl

8. P. Deligne, L. Illusie, Relèvements modulo p ² et décomposition du complexe de de Rham, Invent. Math., 89 (1987), 247-270 | MR | Zbl

9. P. Deligne and J. Milne, Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, Lect. Notes Math., vol. 900, Springer, Berlin Heidelberg New York, 1982. | MR | Zbl

10. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, New York (1999) | MR | Zbl

11. G. Faltings, Crystalline cohomology and p-adic Galois representations, in J.-I. Igusa, ed., Algebraic Analysis, Geometry, and Number Theory, pp. 25-80, The Johns Hopkins University Press, Baltimore London, 1989. | MR | Zbl

12. G. Faltings, Crystalline cohomology of semistable curve - the Q_p-theory, J. Algebr. Geom., 6 (1997), 1-18 | MR | Zbl

13. A. Grothendieck, J. Dieudonné, Elements de géométrie algébrique: étude locale des schémas et des morphismes des schémas, Publ. Math., Inst. Hautes Étud. Sci., 24 (1964), 5-231 | Numdam | Zbl

14. A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique, Grundlehren der mathematischen Wissenschaften, vol. 166, Springer, 1971. | Zbl

15. L. Illusie, Complexe Cotangent et Déformations I, Springer, Berlin Heidelberg New York (1971) | MR | Zbl

16. K. Joshi, C.S. Rajan, Frobenius splitting and ordinarity, Int. Math. Res. Not., 2 (2003), 109-121 | MR | Zbl

17. K. Kato, Logarithmic structures of Fontaine-Illusie, in J.-I. Igusa, ed., Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore London, 1989. | MR | Zbl

18. N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math., Inst. Hautes Étud. Sci., 39 (1970), 175-232 | Numdam | MR | Zbl

19. N. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., 18 (1972), 1-118 | MR | Zbl

20. G. Laumon, Sur la catégorie dérivée des D-modules filtrées, in Algebraic Geometry (Tokyo-Kyoto), pp. 151-237, Springer, Berlin Heidelberg New York, 1983. | MR | Zbl

21. B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc., 78 (1972), 653-667 | MR | Zbl

22. B. Mazur, W. Messing, Universal Extensions and One Dimensional Crystalline Cohomology, Springer, Berlin Heidelberg New York (1974) | MR | Zbl

23. J. Milne, Étale Cohomology, Princeton University Press, Princeton, N.J. (1980) | MR | Zbl

24. A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205-236 | Zbl

25. A. Neeman, Triangulated Categories, Princeton University Press, Princeton, N.J. (2001) | MR | Zbl

26. A. Ogus, F-crystals and Griffiths transversality. in Proceedings of the International Symposium on Algebraic Geometry, Kyoto 1977, pp. 15-44, Kinokuniya Book-Store, Co., Tokyo, 1977. | MR | Zbl

27. A. Ogus, Griffiths transversality in crystalline cohomology, Ann. Math., 108 (1978), 395-419 | MR | Zbl

28. A. Ogus, F-Crystals, Griffiths Transversality, and the Hodge Decomposition, Astérisque, vol. 221, Soc. Math. France, 1994. | MR | Zbl

29. A. Ogus, Higgs cohomology, p-curvature, and the Cartier isomorphism, Compos. Math., 140 (2004), 145-164 | MR | Zbl

30. B. Osserman, Mochizuki's crys-stable bundles: a lexicon and applications, RIMS Kokyuroku, 43 (2007), 95-119

31. M. Raynaud, “p-torsion” du schéma de Picard, Astérisque, 64 (1978), 87-149 | Numdam | Zbl

32. N. S. Rivano, Catégories Tannakiennes, Lect. Notes Math., vol. 265, Springer, 1972. | MR

33. N. Roby, Lois polynômes et lois formelles en théorie des modules, Ann. Éc. Norm. Super., III. Sér., 80 (1963), 213-348 | Numdam | MR | Zbl

34. C. Sabbah, On a twisted de Rham complex, Tohoku Math. J., 51 (1999), 125-140 | MR | Zbl

35. M. Saito, Hodge structure via filtered D-modules, Astérisque, 130 (1985), 342-351 | Numdam | MR | Zbl

36. C. Simpson, Higgs bundles and local systems, Publ. Math., Inst. Hautes Étud. Sci., 75 (1992), 5-95 | Numdam | MR | Zbl

37. V. Srinivas, Decomposition of the de Rham complex, Proc. Indian Acad. Sci., Math. Sci., 100 (1990), 103-106 | MR | Zbl

38. V. Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies, http://www.math.uiuc.edu/K-theory/443, 2000.

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