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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, Mathematical Surveys and Monographs, vol. 33, Am. Math. Soc., Providence, 1990.
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.
S. Bosch and W. Lütkebohmert, Formal rigid geometry. I. Rigid spaces, Math. Ann., 295 (1993), 291–317.
P. Boyer, Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math., 177 (2009), 239–280.
P. Boyer, Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires, Compos. Math., 146 (2010), 367–403.
A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., to appear, arXiv:1010.2188.
J.-F. Dat, Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math., 169 (2007), 75–152.
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math., 83 (1996), 51–93.
P. Deligne, Théorie de Hodge. I, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 425–430, Gauthier-Villars, Paris, 1971.
P. Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252.
G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc., 1 (1988), 255–299.
G. Faltings, Cohomologies p-adiques et applications arithmétiques, II, in Almost étale extensions. Astérisque, vol. 279, pp. 185–270, 2002.
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.
O. Gabber and L. Ramero, Foundations of almost ring theory, http://math.univ-lille1.fr/~ramero/hodge.pdf.
O. Gabber and L. Ramero, Almost Ring Theory, Lecture Notes in Mathematics, vol. 1800, Springer, Berlin, 2003.
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.
E. Hellmann, On arithmetic families of filtered φ-modules and crystalline representations, arXiv:1010.4577, 2011.
M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43–60.
R. Huber, Continuous valuations, Math. Z., 212 (1993), 455–477.
R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z., 217 (1994), 513–551.
R. Huber, A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. Algebraic Geom., 7 (1998), 359–403.
R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, vol. E30, Vieweg, Braunschweig, 1996.
L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239, Springer, Berlin, 1971.
L. Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, vol. 283, Springer, Berlin, 1972.
L. Illusie, Autour du théorème de monodromie locale, in Périodes p-adiques. Astérisque, vol. 223, pp. 9–57, 1994.
T. Ito, Weight-monodromy conjecture for p-adically uniformized varieties, Invent. Math., 159 (2005), 607–656.
T. Ito, Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math., 127 (2005), 647–658.
K. Kedlaya and R. Liu, Relative p-adic Hodge theory, I: Foundations, http://math.mit.edu/~kedlaya/papers/relative-padic-Hodge1.pdf.
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.
M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., 68 (1982), 21–101.
P. Scholze, p-adic Hodge theory for rigid-analytic varieties, arXiv:1205.3463, 2012.
S. W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), 173 (2011), 1645–1741.
J. Tate, Rigid analytic spaces, Invent. Math., 12 (1971), 257–289.
R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., 20 (2007), 467–493.
T. Terasoma, Monodromy weight filtration is independent of ℓ, arXiv:math/9802051, 1998.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Scholze, P. Perfectoid Spaces. Publ.math.IHES 116, 245–313 (2012). https://doi.org/10.1007/s10240-012-0042-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10240-012-0042-x