Given a scheme in characteristic together with a lifting modulo , 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.
@article{PMIHES_2007__106__1_0,
author = {Ogus, A. and Vologodsky, V.},
title = {Nonabelian {Hodge} theory in characteristic $p$},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {1--138},
year = {2007},
publisher = {Springer},
volume = {106},
doi = {10.1007/s10240-007-0010-z},
language = {en},
url = {https://www.numdam.org/articles/10.1007/s10240-007-0010-z/}
}
TY - JOUR AU - Ogus, A. AU - Vologodsky, V. TI - Nonabelian Hodge theory in characteristic $p$ JO - Publications Mathématiques de l'IHÉS PY - 2007 SP - 1 EP - 138 VL - 106 PB - Springer UR - https://www.numdam.org/articles/10.1007/s10240-007-0010-z/ DO - 10.1007/s10240-007-0010-z LA - en ID - PMIHES_2007__106__1_0 ER -
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
1. , 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. | Zbl | MR
2. , , , Faisceaux pervers, Astérisque, 100 (1982), 5-171 | Zbl | MR
3. , , Notes on Crystalline Cohomology, Princeton University Press, Princeton, N.J. (1978) | Zbl | MR
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. , , Geometric Langlands correspondence for -modules in prime characteristic: the Gl(n) case, Pure Appl. Math. Q., 3 (2007), 153-179 | MR
6. , Equations Différentielles à Points Singuliers Réguliers, Springer, Berlin Heidelberg New York (1970) | Zbl | MR
7. , Théorie de Hodge II, Publ. Math., Inst. Hautes Étud. Sci., 40 (1972), 5-57 | Zbl | MR | Numdam
8. , , Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., 89 (1987), 247-270 | Zbl | MR
9. and , Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, Lect. Notes Math., vol. 900, Springer, Berlin Heidelberg New York, 1982. | Zbl | MR
10. , Commutative Algebra with a View Toward Algebraic Geometry, Springer, New York (1999) | Zbl | MR
11. , 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. | Zbl | MR
12. , Crystalline cohomology of semistable curve - the Qp -theory, J. Algebr. Geom., 6 (1997), 1-18 | Zbl | MR
13. , , Elements de géométrie algébrique: étude locale des schémas et des morphismes des schémas, Publ. Math., Inst. Hautes Étud. Sci., 24 (1964), 5-231 | Zbl | Numdam
14. and , Eléments de Géométrie Algébrique, Grundlehren der mathematischen Wissenschaften, vol. 166, Springer, 1971. | Zbl
15. , Complexe Cotangent et Déformations I, Springer, Berlin Heidelberg New York (1971) | Zbl | MR
16. , , Frobenius splitting and ordinarity, Int. Math. Res. Not., 2 (2003), 109-121 | Zbl | MR
17. , Logarithmic structures of Fontaine-Illusie, in J.-I. Igusa, ed., Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore London, 1989. | Zbl | MR
18. , Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math., Inst. Hautes Étud. Sci., 39 (1970), 175-232 | Zbl | MR | Numdam
19. , Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., 18 (1972), 1-118 | Zbl | MR
20. , Sur la catégorie dérivée des D-modules filtrées, in Algebraic Geometry (Tokyo-Kyoto), pp. 151-237, Springer, Berlin Heidelberg New York, 1983. | Zbl | MR
21. , Frobenius and the Hodge filtration, Bull. Amer. Math. Soc., 78 (1972), 653-667 | Zbl | MR
22. , , Universal Extensions and One Dimensional Crystalline Cohomology, Springer, Berlin Heidelberg New York (1974) | Zbl | MR
23. , Étale Cohomology, Princeton University Press, Princeton, N.J. (1980) | Zbl | MR
24. , The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205-236 | Zbl
25. , Triangulated Categories, Princeton University Press, Princeton, N.J. (2001) | Zbl | MR
26. , 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. | Zbl | MR
27. , Griffiths transversality in crystalline cohomology, Ann. Math., 108 (1978), 395-419 | Zbl | MR
28. , F-Crystals, Griffiths Transversality, and the Hodge Decomposition, Astérisque, vol. 221, Soc. Math. France, 1994. | Zbl | MR
29. , Higgs cohomology, p-curvature, and the Cartier isomorphism, Compos. Math., 140 (2004), 145-164 | Zbl | MR
30. , Mochizuki's crys-stable bundles: a lexicon and applications, RIMS Kokyuroku, 43 (2007), 95-119
31. , “p-torsion” du schéma de Picard, Astérisque, 64 (1978), 87-149 | Zbl | Numdam
32. N. S. Rivano, Catégories Tannakiennes, Lect. Notes Math., vol. 265, Springer, 1972. | MR
33. , Lois polynômes et lois formelles en théorie des modules, Ann. Éc. Norm. Super., III. Sér., 80 (1963), 213-348 | Zbl | MR | Numdam
34. , On a twisted de Rham complex, Tohoku Math. J., 51 (1999), 125-140 | Zbl | MR
35. , Hodge structure via filtered D-modules, Astérisque, 130 (1985), 342-351 | Zbl | MR | Numdam
36. , Higgs bundles and local systems, Publ. Math., Inst. Hautes Étud. Sci., 75 (1992), 5-95 | Zbl | MR | Numdam
37. , Decomposition of the de Rham complex, Proc. Indian Acad. Sci., Math. Sci., 100 (1990), 103-106 | Zbl | MR
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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