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.
Accepted:
Online First:
Published online:
DOI: 10.1007/s10240-012-0042-x
Peter Scholze 1
@article{PMIHES_2012__116__245_0,
author = {Peter Scholze},
title = {Perfectoid {Spaces}},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {245--313},
year = {2012},
publisher = {Springer-Verlag},
volume = {116},
doi = {10.1007/s10240-012-0042-x},
zbl = {1263.14022},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0042-x/}
}
TY - JOUR AU - Peter Scholze TI - Perfectoid Spaces JO - Publications Mathématiques de l'IHÉS PY - 2012 SP - 245 EP - 313 VL - 116 PB - Springer-Verlag UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-012-0042-x/ DO - 10.1007/s10240-012-0042-x LA - en ID - PMIHES_2012__116__245_0 ER -
Peter Scholze. Perfectoid Spaces. Publications Mathématiques de l'IHÉS, Volume 116 (2012), pp. 245-313. doi: 10.1007/s10240-012-0042-x
[1.] Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, Mathematical Surveys and Monographs, 33, Am. Math. Soc., Providence, 1990 | Zbl
[2.] Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften, 261, Springer, Berlin, 1984 (A systematic approach to rigid analytic geometry) | Zbl | DOI
[3.] Formal rigid geometry. I. Rigid spaces, Math. Ann., Volume 295 (1993), pp. 291-317 | MR | Zbl | DOI
[4.] Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math., Volume 177 (2009), pp. 239-280 | MR | Zbl | DOI
[5.] Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires, Compos. Math., Volume 146 (2010), pp. 367-403 | MR | Zbl | DOI
[6.] A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., to appear, . | arXiv
[7.] Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math., Volume 169 (2007), pp. 75-152 | MR | Zbl | DOI
[8.] Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math., Volume 83 (1996), pp. 51-93 | DOI | Zbl | Numdam
[9.] Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris (1971), pp. 425-430 | Zbl
[10.] La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math., Volume 52 (1980), pp. 137-252 | DOI | MR | Zbl | Numdam
[11.] p-adic Hodge theory, J. Amer. Math. Soc., Volume 1 (1988), pp. 255-299 | MR | Zbl
[12.] Cohomologies p-adiques et applications arithmétiques, II, Almost étale extensions (Astérisque, 279) (2002), pp. 185-270 | Zbl | Numdam
[13.] Extensions algébrique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A-B, Volume 288 (1979), p. A441-A444 | MR | Zbl
[14.] O. Gabber and L. Ramero, Foundations of almost ring theory, http://math.univ-lille1.fr/~ramero/hodge.pdf. | Zbl
[15.] Almost Ring Theory, Lecture Notes in Mathematics, 1800, Springer, Berlin, 2003 | Zbl
[16.] The Geometry and Cohomology of some Simple Shimura Varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, 2001 (With an appendix by Vladimir G. Berkovich.) | Zbl
[17.] E. Hellmann, On arithmetic families of filtered φ-modules and crystalline representations, , 2011. | arXiv
[18.] Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., Volume 142 (1969), pp. 43-60 | MR | Zbl | DOI
[19.] Continuous valuations, Math. Z., Volume 212 (1993), pp. 455-477 | MR | Zbl | DOI
[20.] A generalization of formal schemes and rigid analytic varieties, Math. Z., Volume 217 (1994), pp. 513-551 | MR | Zbl | DOI
[21.] A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. Algebraic Geom., Volume 7 (1998), pp. 359-403 | MR | Zbl
[22.] Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Vieweg, Braunschweig, 1996 | Zbl
[23.] Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, Berlin, 1971 | Zbl
[24.] Complexe cotangent et déformations. II, Lecture Notes in Mathematics, 283, Springer, Berlin, 1972 | Zbl
[25.] Autour du théorème de monodromie locale, Périodes p-adiques (Astérisque, 223) (1994), pp. 9-57 | Zbl | Numdam
[26.] Weight-monodromy conjecture for p-adically uniformized varieties, Invent. Math., Volume 159 (2005), pp. 607-656 | MR | Zbl | DOI
[27.] Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math., Volume 127 (2005), pp. 647-658 | MR | Zbl | DOI
[28.] K. Kedlaya and R. Liu, Relative p-adic Hodge theory, I: Foundations, http://math.mit.edu/~kedlaya/papers/relative-padic-Hodge1.pdf.
[29.] On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math, XVII), Am. Math. Soc., Providence (1970), pp. 65-87 | Zbl
[30.] Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., Volume 68 (1982), pp. 21-101 | MR | Zbl | DOI
[31.] P. Scholze, p-adic Hodge theory for rigid-analytic varieties, , 2012. | arXiv | Zbl
[32.] Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), Volume 173 (2011), pp. 1645-1741 | MR | Zbl | DOI
[33.] Rigid analytic spaces, Invent. Math., Volume 12 (1971), pp. 257-289 | MR | Zbl | DOI
[34.] Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., Volume 20 (2007), pp. 467-493 | MR | Zbl | DOI
[35.] T. Terasoma, Monodromy weight filtration is independent of ℓ, , 1998. | arXiv
Cited by Sources: