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Dieudonné theory over semiperfect rings and perfectoid rings

Part of: Algebraic groups (Co)homology theory

Published online by Cambridge University Press:  17 August 2018

Eike Lau
Eike Lau*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany email lau@math-uni-bielefeld.de
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Abstract

The Dieudonné crystal of a $p$-divisible group over a semiperfect ring $R$ can be endowed with a window structure. If $R$ satisfies a boundedness condition, this construction gives an equivalence of categories. As an application we obtain a classification of $p$-divisible groups and commutative finite locally free $p$-group schemes over perfectoid rings by Breuil–Kisin–Fargues modules if $p\geqslant 3$.

MSC classification

Primary: 14L05: Formal groups, $p$-divisible groups
Secondary: 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 9 , September 2018 , pp. 1974 - 2004
Copyright
© The Author 2018 

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References

Berthelot, P., Cohomologie cristalline des schémas de charactéristique p > 0, Lecture Notes in Mathematics, vol. 407 (Springer, Berlin–New York, 1974).+0,+Lecture+Notes+in+Mathematics,+vol.+407+(Springer,+Berlin–New+York,+1974).>Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics, vol. 930 (Springer, Berlin, 1982).Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Integral $p$ -adic Hodge theory, Preprint (2016),arXiv:1602.03148 [math.AG].Google Scholar
Bourbaki, N., Algèbre commutative, Éléments de mathématique, vol. 7 (Masson, Paris, 1983), chs 8, 9.Google Scholar
Cais, B. and Lau, E., Dieudonné crystals and Wach modules for $p$ -divisible groups, Preprint (2014), arXiv:1412.3174 [math.NT].Google Scholar
de Jong, J., Finite locally free groups schemes in characteristic p and Dieudonné modules , Invent. Math. 114 (1993), 89–137.Google Scholar
de Jong, J., Crystalline Dieudonné theory via formal and rigid geometry , Publ. Math. Inst. Hautes Études Sci. 82 (1995), 5–96.Google Scholar
Fargues, L., $\unicode[STIX]{x1D711}$ -Modules and modifications of vector bundles, Conference on Arithmetic Algebraic Geometry, Paderborn (2013),http://webusers.imj-prg.fr/∼laurent.fargues/PhiModules_Paderborn.pdf.Google Scholar
Fargues, L., Quelques résultats et conjectures concernant la courbe , Astérisque 369 (2015), 325–374.Google Scholar
Fontaine, J.-M., Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze) , in Séminare Bourbaki, June, 2012, Exp. No. 1057, Astérisque 352 (2013), 509–534.Google Scholar
Gabber, O. and Ramero, L., Foundations for almost ring theory, Release 7,arXiv:math/0409584v12 [math.AG] (2017).Google Scholar
Grothendieck, A., Groupes de Barsotti–Tate et Cristaux de Dieudonné (Université de Montréal, Montréal, QC, 1974).Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique I , Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5–228.Google Scholar
Illusie, L., Deformations de groupes de Barsotti–Tate (d’après A. Grothendieck) , in Seminar on arithmetic bundles: the Mordell conjecture, Astérisque 127 (1985), 151–198.Google Scholar
Kisin, M., Crystalline representations and F-crystals , in Algebraic geometry and number theory, Progress in Mathematics, vol. 253 (Birkhäuser, Boston, MA, 2006), 459–496.Google Scholar
Lau, E., Displays and formal p-divisible groups , Invent. Math. 171 (2008), 617–628.Google Scholar
Lau, E., Frames and finite group schemes over complete regular local rings , Doc. Math. 15 (2010), 545–569.Google Scholar
Lau, E., Smoothness of the truncated display functor , J. Amer. Math. Soc. 26 (2013), 129–165.Google Scholar
Lau, E., Relations between crystalline Dieudonné theory and Dieudonné displays , Algebra Number Theory 8 (2014), 2201–2262.Google Scholar
Mazur, B. and Messing, W., Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics, vol. 370 (Springer, Berlin–New York, 1974).Google Scholar
Messing, W., The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 264 (Springer, Berlin–New York, 1972).Google Scholar
Scholze, P. and Weinstein, J., Moduli spaces of p-divisible groups , Camb. J. Math. 1 (2013), 145–237.Google Scholar
Scholze, P. and Weinstein, J., Berkeley lectures on p-adic geometry (2017), http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf.Google Scholar
The Stacks Project Authors: Stacks Project (2016), http://stacks.math.columbia.edu.Google Scholar
Zink, T., Windows for displays of p-divisible groups. Moduli of abelian varieties, Progress in Mathematics, vol. 195 (Birkhäuser, Basel, 2001), 491–518.Google Scholar
Zink, T., The display of a formal p-divisible group , in Cohomologies p-adiques et applications arithmétiques, I, Astérisque 278 (2002), 127–248.Google Scholar