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Smoothness of the truncated display functor


Author: Eike Lau
Journal: J. Amer. Math. Soc. 26 (2013), 129-165
MSC (2010): Primary 14F30, 14L05
DOI: https://doi.org/10.1090/S0894-0347-2012-00744-9
Published electronically: June 28, 2012
MathSciNet review: 2983008
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Abstract: We show that to every $ p$-divisible group over a $ p$-adic ring one can associate a display by crystalline Dieudonné theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal $ p$-divisible groups and nilpotent displays over $ p$-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of $ p$-power order and Dieudonné modules over perfect rings.


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  • [Be] P. Berthelot: Théorie de Dieudonné sur un anneau de valuation parfait. Ann. Sci. Ec. Norm. Sup. (4) 13 (1980), 225-268 MR 584086 (82b:14026)
  • [BBM] P. Berthelot, L. Breen, and W. Messing: Théorie de Dieudonné cristalline II. Lecture Notes in Math. 930, Springer-Verlag, 1982 MR 667344 (85k:14023)
  • [BM] P. Berthelot and W. Messing: Théorie de Dieudonné cristalline, III. The Grothendieck Festschrift, Vol. I, 173-247, Progr. Math. 86, Birkhäuser, 1990 MR 1086886 (92h:14012)
  • [Bü] O. Bültel: PEL modulispaces without $ \mathbb{C}$-valued points. arxiv.org:0808.4091.
  • [G1] A. Grothendieck: Groupes de Barsotti-Tate et cristaux. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 431-436. Gauthier-Villars, Paris, 1971 MR 0578496 (58:28211)
  • [G2] A. Grothendieck: Groupes de Barsotti-Tate et Cristaux de Dieudonné. Université de Montréal, 1974 MR 0417192 (54:5250)
  • [Il1] L. Illusie: Complexe cotangent et déformations, I. Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 MR 0491680 (58:10886a)
  • [Il2] L. Illusie: Deformations de groupes de Barsotti-Tate (d'après A. Grothendieck). In Seminar on arithmetic bundles: the Mordell conjecture, Astérisque 127 (1985), 151-198 MR 801922
  • [Ka] N. M. Katz: Slope filtration of $ F$-crystals. Journées de Géométrie Algébrique de Rennes, Astérisque 63 (1979), 113-163 MR 563463 (81i:14014)
  • [Ki] M. Kisin: Crystalline representations and $ F$-crystals, Algebraic geometry and number theory, 459-496, Progr. Math., Vol. 253, Birkhäuser, 2006 MR 2263197 (2007j:11163)
  • [LZ] A. Langer and Th. Zink: De Rham-Witt cohomology for a proper and smooth morphism. J. Inst. Math. Jussieu 3 (2004), no. 2, 231-314 MR 2055710 (2005d:14027)
  • [La1] E. Lau: Displays and formal $ p$-divisible groups. Invent. Math. 171 (2008), 617-628 MR 2372808 (2009j:14058)
  • [La2] E. Lau: Frames and finite group schemes over complete regular local rings. Doc. Math. 15 (2010), 545-569. MR 2679066 (2011g:14107)
  • [La3] E. Lau: Relations between crystalline Dieudonné theory and Dieudonné displays. arxiv.org:1006.2720
  • [Ma] H. Matsumura: Commutative ring theory. Cambridge Univ. Press, 1986 MR 879273 (88h:13001)
  • [Me1] W. Messing: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, 1972 MR 0347836 (50:337)
  • [Me2] W. Messing: Travaux de Zink. Séminaire Bourbaki 2005/2006, exp. 964, Astérisque 311 (2007), 341-364 MR 2359049 (2009c:14088)
  • [NVW] M.-H. Nicole, A. Vasiu, and T. Wedhorn, Purity of level $ m$ stratifications. Ann. Sci. Ec. Norm. Sup. (4) 43 (2010), 925-955. MR 2778452 (2012a:14100)
  • [O1] F. Oort: Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. of Math. (2) 152 (2000), 183-206 MR 1792294 (2002e:14075)
  • [Po] D. Popescu: General Néron desingularization and approximation. Nagoya Math. J. 104 (1986), 85-115 MR 868439 (88a:14007)
  • [RZ] M. Rapoport and Th. Zink: Period spaces of $ p$-divisible groups. Ann. Math. Stud. 141, Princeton Univ. Press, 1996 MR 1393439 (97f:14023)
  • [Sw] R: Swan: Néron-Popescu desingularization. In: Algebra and geometry (Taipei, 1995), 135-192, Lect. Algebra Geom. 2, Int. Press, Cambridge, MA, 1998 MR 1697953 (2000h:13006)
  • [Va] P. Valabrega: A few theorems on completion of excellent rings. Nagoya Math. J. 61 (1976), 127-133 MR 0407007 (53:10790)
  • [W] T. Wedhorn: The dimension of Oort strata of Shimura varieties of PEL-type. In: Moduli of Abelian Varieties, Progr. Math. Vol. 195, 441-471, Birkhäuser, Basel, 2001 MR 1827029 (2002b:14029)
  • [Zi1] Th. Zink: The display of a formal $ p$-divisible group. In: Cohomologies $ p$-adiques et applications arithmétiques, I, Astérisque 278 (2002), 127-248 MR 1922825 (2004b:14083)

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Additional Information

Eike Lau
Affiliation: Institut für Mathematik, Universität Paderborn, D-33098 Paderborn, Germany
Email: elau@math.upb.de

DOI: https://doi.org/10.1090/S0894-0347-2012-00744-9
Received by editor(s): September 5, 2011
Received by editor(s) in revised form: May 25, 2012
Published electronically: June 28, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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