PD Higgs crystals and Higgs cohomology in characteristic $p$
Author:
Hidetoshi Oyama
Journal:
J. Algebraic Geom. 26 (2017), 735-802
DOI:
https://doi.org/10.1090/jag/699
Published electronically:
May 26, 2017
MathSciNet review:
3683425
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Abstract |
References |
Additional Information
Abstract: For a smooth scheme $X$ over a perfect field of characteristic $p$, we define two sites and show that if a lifting of $X$ over the ring of Witt vectors is given, the categories of crystals on these sites are equivalent to the categories of certain $\mathcal {D}$-modules and certain Higgs modules respectively. We construct a natural functor between these sites and show that the morphism of topoi associated to this functor induces an equivalence between the categories of quasi-coherent crystals on these sites. We also study a filtered version of these results and give interpretations of the de Rham cohomology and the Higgs cohomology in terms of filtered crystals. We use it to give an alternative proof of the cohomology comparison theorem due to Ogus and Vologodsky.
References
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References
- Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458
- Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). MR 0354652
- Pierre Berthelot, Lawrence Breen, and William Messing, Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982 (French). MR 667344
- Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). MR 0384804
- Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 0491705
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 0291177
- Arthur Ogus, Higgs cohomology, $p$-curvature, and the Cartier isomorphism, Compos. Math. 140 (2004), no. 1, 145–164. MR 2004127, DOI https://doi.org/10.1112/S0010437X03000058
- A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic $p$, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138. MR 2373230, DOI https://doi.org/10.1007/s10240-007-0010-z
- Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076
- Stacks Project, stacks.math.columbia.edu.
- Takeshi Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), no. 2, 233–411. MR 1705837, DOI https://doi.org/10.1007/s002220050330
- Takeshi Tsuji, On nearby cycles and $\mathcal {D}$-modules of log schemes in characteristic $p>0$, Compos. Math. 146 (2010), no. 6, 1552–1616. MR 2735373, DOI https://doi.org/10.1112/%5Cspace%20S0010437X10004768
- Takeshi Tsuji, Cohomology of Higgs isocrystals, The $p$-adic Simpson correspondence, Ann. of Math. Stud., vol. 193, Princeton Univ. Press, Princeton, NJ, 2016, pp. 307–447. MR 3444781
Additional Information
Hidetoshi Oyama
Email:
hidetoshi1129@gmail.com
Received by editor(s):
August 10, 2015
Received by editor(s) in revised form:
August 31, 2016
Published electronically:
May 26, 2017
Article copyright:
© Copyright 2017
University Press, Inc.