Crystalline Dieudonné module theory via formal and rigid geometry. (English) Zbl 0864.14009

Dieudonné theory associates to a \(p\)-adic divisible group a Dieudonné-crystal, that is (roughly) a module with connection, fibration and Frobenius. The idea is that these are easier to handle than \(p\)-divisible groups. Except for some minor technical restrictions the paper shows that over a field \(R\) of characteristic \(p\) the functor is an equivalence of categories for smooth formal schemes, and still fully faithful up to isogeny for arbitrary reduced schemes. For the first result the author first treats fields, then studies deformations, and finally uses noetherian induction. For this last step one is already forced to work in the additional generality of formal schemes. The proof of the second result is very complicated and uses rigid-analytic methods.
Reviewer: G.Faltings (Bonn)


14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
Full Text: DOI Numdam Numdam EuDML


[1] W. Bartenwerfer, Einige Fortsetzungssätze in derp-adischen Analysis,Math. Ann.,185 (1970), 191–210. · doi:10.1007/BF01350260
[2] P. Berthelot,Cohomologie cristalline des schémas de charactéristique p>0, Springer Lecture Notes,407, 1974. · Zbl 0298.14012
[3] P. Berthelot,Cohomologie rigide et cohomologie rigide à support propre, Version privisoire du 9-08-1991. · Zbl 0515.14015
[4] P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristiquep, Bull. Soc. Math. Fr., Mém.,23 (1986), 7–32.
[5] P. Berthelot, L. Breen, W. Messing,Théorie de Dieudonné cristalline II, Springer Lecture Notes,930, 1982.
[6] P. Berthelot, W. Messing, Théorie de Dieudonné cristalline III, inThe Grothendieck Festschrift I, Progress in Mathematics,86, Birkhäuser (1990), 171–247.
[7] S. Bosch, U. Güntzer, R. Remmert,Non-archimedian analysis, Grundlehren,261, Springer Verlag, 1984. · Zbl 0539.14017
[8] S. Bosch, W. Lütkebohmert, Formal and rigid geometry,Math. Ann.,295 (1993), 291–317. · Zbl 0808.14017 · doi:10.1007/BF01444889
[9] A. Grothendieck,Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de mathématiques supérieures, Département de mathématiques, Université de Montréal, Les Presses de l’Université de Montréal, 1974.
[10] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné,Actes, Congrès international des mathématiciens, 1970, t. 1, p. 431–436.
[11] A. Grothendieck, J. Dieudonné,Éléments de géométrie algébrique I, II, III, IV, Publ. Math. I.H.E.S.,4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).
[12] R. Hartshorne, Complete intersections and connectedness,Amer. J. Math.,84 (1962), 495–508. · Zbl 0108.16602 · doi:10.2307/2372986
[13] L. Illusie,Complexe cotangent et déformations I, II, Springer Lecture Notes,239 (1971),283 (1972). · Zbl 0224.13014
[14] L. Illusie, Déformations de groupes de Barsotti-Tate, d’après A. Grothendieck, inSéminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque,127 (1985), 151–198. · Zbl 1182.14050
[15] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline,Ann. Scient. Éc. Norm. Sup.,12 (1979), 501–661. · Zbl 0436.14007
[16] A. J. de Jong, Finite locally free group schemes in characteristicp and Dieudonné modules,Invent. Math.,114 (1993), 89–137. · Zbl 0812.14030 · doi:10.1007/BF01232664
[17] A. J. deJong, M. van der Put,Étale cohomology of rigid analytic spaces, Preprint of the University of Groningen, W-9506. · Zbl 0922.14012
[18] J.-P. Jouanolou,Théorèmes de Bertini et applications, Progress in Mathematics42, Birkhäuser, 1983.
[19] N. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turittin,Publ. Math. I.H.E.S.,35 (1971), 175–232.
[20] J. Lipman, Desingularization of two-dimensional schemes,Ann. of Math.,107 (1978), 151–207. · doi:10.2307/1971141
[21] W. Lütkebohmert, Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie,Math. Z.,139 (1974), 69–84. · Zbl 0283.32022 · doi:10.1007/BF01194146
[22] W. Lütkebohmert, Fortsetzbarkeitk-meromorpher Funktionen,Math. Ann.,220 (1976), 273–284. · Zbl 0319.32025 · doi:10.1007/BF01431097
[23] H. Matsumura,Commutative algebra, Mathematics lecture note series,56, The Benjamin/Cummings Publishing Company, Inc., 1980. · Zbl 0441.13001
[24] B. Mazur, W. Messing,Universal extensions and one-dimensional crystalline cohomology, Springer Lecture Notes,370 (1974). · Zbl 0301.14016
[25] W. Messing,The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Springer Lecture Notes,264 (1972). · Zbl 0243.14013
[26] M. Raynaud, Géométrie analytique rigide d’après Tate, Kiehl,Table ronde d’analyse non archimédienne, Bull. Soc. Math. Fr. Mém.,39/40 (1974), 319–327.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.