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$p$-ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

Part of: Arithmetic problems. Diophantine geometry Cycles and subschemes Compact analytic spaces

Published online by Cambridge University Press:  17 May 2013

PETER SCHOLZE
PETER SCHOLZE*
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, [email protected]

Abstract

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We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.

MSC classification

Secondary: 14G22: Rigid analytic geometry 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14G20: Local ground fields 32J27: Compact Kähler manifolds
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 1 , 2013 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

References

Andreatta, F. and Iovita, A., ‘Comparison isomorphisms for smooth formal schemes’, J. Inst. Math. Jussieu 12(1) (2013), 77–151.Google Scholar
Beilinson, A., ‘ $p$ -adic periods and derived de Rham cohomology’, J. Amer. Math. Soc. 25 (2012), 715–738.Google Scholar
Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean Analysis: a Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261 (Springer, Berlin, 1984).Google Scholar
Brinon, O., ‘Représentations $p$ -adiques cristallines et de de Rham dans le cas relatif’, Mém. Soc. Math. Fr. (N.S.) 112 (2008), vi+159.Google Scholar
Colmez, P., ‘Espaces de Banach de dimension finie’, J. Inst. Math. Jussieu 1(3) (2002), 331–439.Google Scholar
Faltings, G., ‘Almost étale extensions’, Astérisque 279 (2002), 185–270, cohomologies $p$ -adiques et applications arithmétiques, II.Google Scholar
Gabber, O. and Ramero, L., Foundations of Almost Ring Theory. http://math.univ-lille1.fr/~ramero/hodge.pdf.Google Scholar
Gabber, O. and Ramero, L., Almost Ring Theory, Lecture Notes in Mathematics, 1800 (Springer, Berlin, 2003).Google Scholar
Huber, R., Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects of Mathematics, E30 (Friedr. Vieweg & Sohn, Braunschweig, 1996).Google Scholar
de Jong, A. J., ‘Étale fundamental groups of non-Archimedean analytic spaces’, Compositio Math. 97(1–2) (1995), 89–118, special issue in honour of Frans Oort.Google Scholar
de Jong, J. and van der Put, M., ‘Étale cohomology of rigid analytic spaces’, Doc. Math. 1(01) (1996), 1–56 (electronic).Google Scholar
Kiehl, R., ‘Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie’, Invent. Math. 2 (1967), 191–214.Google Scholar
Köpf, U., ‘Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen’, Schr. Math. Inst. Univ. Münster (2) (Heft 7) (1974), iv+72.Google Scholar
Lütkebohmert, W., ‘From Tate’s elliptic curve to abeloid varieties’, Pure Appl. Math. Q. 5(4, Special Issue: In honor of John Tate. Part 1),  (2009), 1385–1427.Google Scholar
Rapoport, M. and Zink, T., Period Spaces for p-divisible Groups, Annals of Mathematics Studies, 141 (Princeton University Press, Princeton, NJ, 1996).Google Scholar
Schneider, P., ‘The cohomology of local systems on $p$ -adically uniformized varieties’, Math. Ann. 293(4) (1992), 623–650.Google Scholar
Scholze, P., ‘Perfectoid spaces’, Publ. Math. Inst. Hautes Études Sci. 116(1) (2012), 245–313.Google Scholar
Tate, J. T., ‘ $p$ -divisible groups’, Proc. Conf. Local Fields (Driebergen, 1966) (Springer, Berlin, 1967), 158–183.Google Scholar
Temkin, M., ‘On local properties of non-Archimedean analytic spaces’, Math. Ann. 318(3) (2000), 585–607.Google Scholar