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Crystalline Dieudonné theory over excellent schemes. (English) Zbl 0963.14008

Let \(\mathbb{D}\) stand for the crystalline Dieudonné module functor on \(p\)-divisible groups over a base \(S\) of characteristic \(p\). The main results are: the full-faithfulness of \(\mathbb{D}\) over excellent local complete intersection schemes, and the full-faithfulness of \(\mathbb{D}\) up to isogeny when \(S\) is local excellent.
The authors make use of D. Popescu’s strong Néron desingularization [D. Popescu, Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008)] and the extension theorem for homomorphisms of \(p\)-divisible groups of A. J. de Jong [Invent. Math. 134, No. 2, 301-333 (1998; Zbl 0929.14029); Erratum: ibid. 138, No. 1, 225 (1999)]. They also prove results concerning the faithfulness for the Dieudonné functor on finite locally free group schemes (there are examples where it is not faithful).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups

References:

[1] BERTHELOT (P.) . - Cohomologie rigide et cohomologie rigide à support propre , version provisoire du 9-08- 1991 . · Zbl 0515.14015
[2] BERTHELOT (P.) , BREEN (L.) , MESSING (W.) . - Théorie de Dieudonné cristalline , II, Lecture Notes in Mathematics 930, Springer Verlag, 1982 . MR 85k:14023 | Zbl 0516.14015 · Zbl 0516.14015
[3] BERTHELOT (P.) , MESSING (W.) . - Théorie de Dieudonné cristalline , III, in The Grothendieck Festschrift I, Progress in Math. 86, Birkhäuser, 1990 , p. 171-247. MR 92h:14012 | Zbl 0753.14041 · Zbl 0753.14041
[4] DE JONG (A.J.) . - Crystalline Dieudonné module theory via formal and rigid geometry , Publications Math., t. 82, 1995 , p. 5-96. Numdam | MR 97f:14047 | Zbl 0864.14009 · Zbl 0864.14009 · doi:10.1007/BF02698637
[5] DE JONG (A.J.) . - Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic , Invent. Math., t. 134, 1998 . MR 2000f:14070a | Zbl 0929.14029 · Zbl 0929.14029 · doi:10.1007/s002220050266
[6] KIMURA (T.) , NIITSUMA (N.) . - Regular local rings of characteristic p and p-basis , J. Math. Soc. Japan, t. 32, 1980 , p. 363-371. Article | MR 81j:13022 | Zbl 0451.13010 · Zbl 0451.13010 · doi:10.2969/jmsj/03220363
[7] OORT (F.) . - Commutative group schemes , Lecture Notes in Mathematics 15, Springer, 1966 . MR 35 #4229 | Zbl 0216.05603 · Zbl 0216.05603 · doi:10.1007/BFb0097479
[8] POPESCU (D.) . - General Néron desingularization , Nagoya Math. J., t. 100, 1985 , p. 97-126. Article | MR 87f:13019 | Zbl 0561.14008 · Zbl 0561.14008
[9] SWAN (R.G.) . - Néron-Popescu desingularization , Lecture Notes of the University of Chicago, 1995 ; to appear in Proceedings of the International Conference on Algebra and Geometry, Taipei (Taiwan, 1995 ), p. 1-44.
[10] DEMAZURE (M.) , GROTHENDIECK (A.) . - SGA3, Schémas en groupes , I, Lecture Notes in Math. 151, Springer Verlag, 1970 . MR 43 #223a | Zbl 0207.51401 · Zbl 0207.51401 · doi:10.1007/BFb0058993
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