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Perfectoid Spaces

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Abstract

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic.

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References

  1. V. G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, Mathematical Surveys and Monographs, vol. 33, Am. Math. Soc., Providence, 1990.

    MATH  Google Scholar 

  2. S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer, Berlin, 1984. A systematic approach to rigid analytic geometry.

    Book  MATH  Google Scholar 

  3. S. Bosch and W. Lütkebohmert, Formal rigid geometry. I. Rigid spaces, Math. Ann., 295 (1993), 291–317.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Boyer, Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math., 177 (2009), 239–280.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Boyer, Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires, Compos. Math., 146 (2010), 367–403.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., to appear, arXiv:1010.2188.

  7. J.-F. Dat, Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math., 169 (2007), 75–152.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math., 83 (1996), 51–93.

    Article  MATH  Google Scholar 

  9. P. Deligne, Théorie de Hodge. I, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 425–430, Gauthier-Villars, Paris, 1971.

    Google Scholar 

  10. P. Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc., 1 (1988), 255–299.

    MathSciNet  MATH  Google Scholar 

  12. G. Faltings, Cohomologies p-adiques et applications arithmétiques, II, in Almost étale extensions. Astérisque, vol. 279, pp. 185–270, 2002.

    Google Scholar 

  13. J.-M. Fontaine and J.-P. Wintenberger, Extensions algébrique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A441–A444.

    MathSciNet  Google Scholar 

  14. O. Gabber and L. Ramero, Foundations of almost ring theory, http://math.univ-lille1.fr/~ramero/hodge.pdf.

  15. O. Gabber and L. Ramero, Almost Ring Theory, Lecture Notes in Mathematics, vol. 1800, Springer, Berlin, 2003.

    MATH  Google Scholar 

  16. M. Harris and R. Taylor, The Geometry and Cohomology of some Simple Shimura Varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, 2001. With an appendix by Vladimir G. Berkovich.

    MATH  Google Scholar 

  17. E. Hellmann, On arithmetic families of filtered φ-modules and crystalline representations, arXiv:1010.4577, 2011.

  18. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43–60.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Huber, Continuous valuations, Math. Z., 212 (1993), 455–477.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z., 217 (1994), 513–551.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Huber, A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. Algebraic Geom., 7 (1998), 359–403.

    MathSciNet  MATH  Google Scholar 

  22. R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, vol. E30, Vieweg, Braunschweig, 1996.

    MATH  Google Scholar 

  23. L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239, Springer, Berlin, 1971.

    MATH  Google Scholar 

  24. L. Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, vol. 283, Springer, Berlin, 1972.

    MATH  Google Scholar 

  25. L. Illusie, Autour du théorème de monodromie locale, in Périodes p-adiques. Astérisque, vol. 223, pp. 9–57, 1994.

    Google Scholar 

  26. T. Ito, Weight-monodromy conjecture for p-adically uniformized varieties, Invent. Math., 159 (2005), 607–656.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Ito, Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math., 127 (2005), 647–658.

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Kedlaya and R. Liu, Relative p-adic Hodge theory, I: Foundations, http://math.mit.edu/~kedlaya/papers/relative-padic-Hodge1.pdf.

  29. D. Quillen, On the (co-) homology of commutative rings, in Applications of Categorical Algebra, Proc. Sympos. Pure Math, vol. XVII, pp. 65–87, Am. Math. Soc., Providence, 1970.

    Google Scholar 

  30. M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., 68 (1982), 21–101.

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Scholze, p-adic Hodge theory for rigid-analytic varieties, arXiv:1205.3463, 2012.

  32. S. W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), 173 (2011), 1645–1741.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Tate, Rigid analytic spaces, Invent. Math., 12 (1971), 257–289.

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., 20 (2007), 467–493.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Terasoma, Monodromy weight filtration is independent of ℓ, arXiv:math/9802051, 1998.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, 53115, Bonn, Germany

    Peter Scholze

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  1. Peter Scholze
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Correspondence to Peter Scholze.

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Scholze, P. Perfectoid Spaces. Publ.math.IHES 116, 245–313 (2012). https://doi.org/10.1007/s10240-012-0042-x

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  • DOI: https://doi.org/10.1007/s10240-012-0042-x

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