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Publications mathématiques de l'IHÉS

, Volume 128, Issue 1, pp 219–397 | Cite as

Integral \(p\)-adic Hodge theory

  • Bhargav BhattEmail author
  • Matthew Morrow
  • Peter Scholze
Article
  • 79 Downloads

Abstract

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of \(\mathbf {C}_{p}\). It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known \(p\)-adic cohomology theories, such as crystalline, de Rham and étale cohomology, which allows us to prove strong integral comparison theorems.

The construction of the cohomology theory relies on Faltings’ almost purity theorem, along with a certain functor \(L\eta \) on the derived category, defined previously by Berthelot–Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham–Witt complexes of Langer–Zink, and can be computed as a \(q\)-deformation of de Rham cohomology.

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References

  1. 1.
    The Stacks Project, available at http://stacks.math.columbia.edu.
  2. 2.
    A. Abbes and M. Gros, Topos co-évanescents et généralisations, Annals of Mathematics Studies, vol. 193, 2015. Google Scholar
  3. 3.
    F. Andreatta and A. Iovita, Comparison isomorphisms for smooth formal schemes, J. Inst. Math. Jussieu, 12 (2013), 77–151. MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Beauville and Y. Laszlo, Un lemme de descente, C. R. Acad. Sci., Sér. 1 Math., 320 (1995), 335–340. MathSciNetzbMATHGoogle Scholar
  5. 5.
    P. Berthelot, Sur le “théorème de Lefschetz faible” en cohomologie cristalline, C. R. Acad. Sci. Paris, Sér. A-B, 277 (1973), A955–A958. zbMATHGoogle Scholar
  6. 6.
    P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton University Press/University of Tokyo Press, Princeton/Tokyo, 1978. zbMATHGoogle Scholar
  7. 7.
    P. Berthelot and A. Ogus, \(F\)-isocrystals and de Rham cohomology. I, Invent. Math., 72 (1983), 159–199. MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Bhatt, M. Morrow and P. Scholze, Topological Hochschild homology and integral \(p\)-adic Hodge theory, available at arXiv:1802.03261 [math.AG].
  9. 9.
    B. Bhatt, M. Morrow and P. Scholze, Integral \(p\)-adic Hodge theory—announcement, Math. Res. Lett., 22 (2015), 1601–1612. MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    B. Bhatt and P. Scholze, Prisms and prismatic cohomology, in preparation. Google Scholar
  11. 11.
    B. Bhatt and P. Scholze, The pro-étale topology for schemes, Astérisque, 369 (2015), 99–201. zbMATHGoogle Scholar
  12. 12.
    S. Bloch and K. Kato, \(p\)-adic étale cohomology, Publ. Math. IHÉS, 63 (1986), 107–152. CrossRefzbMATHGoogle Scholar
  13. 13.
    E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. \(p\). III, Invent. Math., 35 (1976), 197–232. MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Borger, The basic geometry of Witt vectors, I: The affine case, Algebra Number Theory, 5 (2011), 231–285. MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer, Berlin, 1984, A systematic approach to rigid analytic geometry. zbMATHGoogle Scholar
  16. 16.
    S. Bosch and W. Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann., 295 (1993), 291–317. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C. Breuil, Groupes \(p\)-divisibles, groupes finis et modules filtrés, Ann. Math., 152 (2000), 489–549. MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    O. Brinon, Représentations \(p\)-adiques cristallines et de de Rham dans le cas relatif, in Mém. Soc. Math. Fr. (N. S.), vol. 112, 2008, vi+159 pp. Google Scholar
  19. 19.
    X. Caruso, Conjecture de l’inertie modérée de Serre, Invent. Math., 171 (2008), 629–699. MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. Colmez and W. Nizioł, Syntomic complexes and \(p\)-adic nearby cycles, Invent. Math., 208 (2017), 1–108. MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    B. Conrad and O. Gabber, Spreading out of rigid-analytic varieties, in preparation. Google Scholar
  22. 22.
    C. Davis and K. S. Kedlaya, On the Witt vector Frobenius, Proc. Am. Math. Soc., 142 (2014), 2211–2226. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. IHÉS, 83 (1996), 51–93. MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Deligne and L. Illusie, Relèvements modulo \(p^{2}\) et décomposition du complexe de de Rham, Invent. Math., 89 (1987), 247–270. MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
  26. 26.
    R. Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. Éc. Norm. Supér., 6 (1973), 553–603. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    G. Faltings, \(p\)-adic Hodge theory, J. Am. Math. Soc., 1 (1988), 255–299. MathSciNetzbMATHGoogle Scholar
  28. 28.
    G. Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Am. Math. Soc., 12 (1999), 117–144. MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Faltings, Almost étale extensions, Cohomologies \(p\)-adiques et applications arithmétiques, II, Astérisque, 279 (2002), 185–270. zbMATHGoogle Scholar
  30. 30.
    L. Fargues, Quelques résultats et conjectures concernant la courbe, Astérisque, 369 (2015), 325–374. MathSciNetzbMATHGoogle Scholar
  31. 31.
    L. Fargues and J.-M. Fontaine, Courbes et fibrés vectoriels en théorie de Hodge \(p\)-adique, available at http://webusers.imj-prg.fr/~laurent.fargues/Courbe_fichier_principal.pdf.
  32. 32.
    J.-M. Fontaine, Sur certains types de représentations \(p\)-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. Math., 115 (1982), 529–577. MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J.-M. Fontaine, Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze), Astérisque, 352 (2013), 509–534, Exp. No. 1057, x. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058. zbMATHGoogle Scholar
  34. 34.
    J.-M. Fontaine and W. Messing, \(p\)-adic periods and \(p\)-adic étale cohomology, in Current Trends in Arithmetical Algebraic Geometry, Arcata, Calif., 1985, Contemp. Math., vol. 67, pp. 179–207, Am. Math. Soc., Providence, 1987. CrossRefGoogle Scholar
  35. 35.
    J.-M. Fontaine and Y. Ouyang, Theory of \(p\)-adic Galois representations, available at https://www.math.u-psud.fr/~fontaine/galoisrep.pdf.
  36. 36.
    O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal., 11 (2001), 1192–1200. MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    O. Gabber, L. Ramero, Almost Ring Theory, Lecture Notes in Mathematics, vol. 1800, Springer, Berlin, 2003, vi+307 pp. ISBN 3-540-40594-1. zbMATHGoogle Scholar
  38. 38.
    O. Gabber and L. Ramero, Foundations of almost ring theory, http://math.univ-lille1.fr/~ramero/hodge.pdf.
  39. 39.
    T. Geisser and L. Hesselholt, The de Rham-Witt complex and \(p\)-adic vanishing cycles, J. Am. Math. Soc., 19 (2006), 1–36. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    L. Hesselholt, On the topological cyclic homology of the algebraic closure of a local field, in An Alpine Anthology of Homotopy Theory, Contemp. Math., vol. 399, pp. 133–162, Am. Math. Soc., Providence, 2006. CrossRefGoogle Scholar
  41. 41.
    R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z., 217 (1994), 513–551. MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects of Mathematics, vol. E30, Vieweg, Braunschweig, 1996. CrossRefzbMATHGoogle Scholar
  43. 43.
    L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér., 12 (1979), 501–661. CrossRefzbMATHGoogle Scholar
  44. 44.
    L. Illusie and M. Raynaud, Les suites spectrales associées au complexe de de Rham-Witt, Publ. Math. IHÉS, 57 (1983), 73–212. CrossRefzbMATHGoogle Scholar
  45. 45.
    N. M. Katz, \(p\)-adic properties of modular schemes and modular forms, in Modular Functions of One Variable III, Lecture Notes in Mathematics, vol. 350, pp. 69–190, 1973. CrossRefGoogle Scholar
  46. 46.
    K. S. Kedlaya, Nonarchimedean geometry of Witt vectors, Nagoya Math. J., 209 (2013), 111–165. MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    K. S. Kedlaya, Some ring-theoretic properties of A_inf, arXiv:1602.09016.
  48. 48.
    K. S. Kedlaya and R. Liu, Relative \(p\)-adic Hodge theory: foundations, Astérisque, 371 (2015), 239. MathSciNetzbMATHGoogle Scholar
  49. 49.
    M. Kisin, Crystalline representations and \(F\)-crystals, in Algebraic Geometry and Number Theory, Progr. Math., vol. 253, pp. 459–496, Birkhäuser Boston, Boston, 2006. CrossRefGoogle Scholar
  50. 50.
    M. Kisin, Integral models for Shimura varieties of Abelian type, J. Am. Math. Soc., 23 (2010), 967–1012. MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    W. E. Lang, On Enriques surfaces in characteristic \(p\). I, Math. Ann., 265 (1983), 45–65. MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    A. Langer and T. Zink, De Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu, 3 (2004), 231–314. MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    C. Liedtke, Arithmetic moduli and lifting of Enriques surfaces, J. Reine Angew. Math., 706 (2015), 35–65. MathSciNetzbMATHGoogle Scholar
  54. 54.
    W. Lütkebohmert, Formal-algebraic and rigid-analytic geometry, Math. Ann., 286 (1990), 341–371. MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    B. Poonen, Bertini theorems over finite fields, Ann. Math., 160 (2004), 1099–1127. MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., 13 (1971), 1–89. MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    M. Schlessinger, Functors of Artin rings, Trans. Am. Math. Soc., 130 (1968), 208–222. MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., 116 (2012), 245–313. MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    P. Scholze, \(p\)-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi, 1, e1, 77 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    P. Scholze, Perfectoid spaces: a survey, in Current Developments in Mathematics 2012, pp. 193–227, International Press, Somerville, 2013. Google Scholar
  61. 61.
    P. Scholze, \(p\)-adic Hodge theory for rigid-analytic varieties—corrigendum [MR3090230], Forum Math. Pi, 4, e6, 4 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    P. Scholze and J. Weinstein, \(p\)-adic geometry, Lecture notes from course at UC Berkeley in Fall 2014, available at https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf.
  63. 63.
    F. Tan and J. Tong, Crystalline comparison isomorphisms in p-adic hodge theory: the absolutely unramified case, available at http://arxiv.org/abs/1510.05543.
  64. 64.
    T. Tsuji, \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math., 137 (1999), 233–411. MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    W. van der Kallen, Descent for the \(K\)-theory of polynomial rings, Math. Z., 191 (1986), 405–415. MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.CNRS and IMJ-PRGSorbonne UniversityParisFrance
  3. 3.Mathematisches InstitutUniversität BonnBonnGermany

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