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Finite locally free group schemes in characteristic \(p\) and Dieudonné modules. (English) Zbl 0812.14030
The general theme of this well written article is the analogue of classical Dieudonné theory for commutative finite locally free \(p\)- group schemes over a scheme \(S_ 0\) of characteristic \(p\), i.e. the description of such groups by objects generalizing Dieudonné modules.
The setting is the following: \(S\) is a flat scheme over \(\mathbb{Z}/p^{n+1} \mathbb{Z}\) \((n \geq 1)\), \(S_ 0 \subset S\) the subscheme where \(p = 0\), \(C(n)_{S_ 0}\) the category of commutative finite locally free group schemes over \(S_ 0\) on which \(p^ n\) is zero. The paper studies properties of the functor \(M_ S\) which to an object \(G\) of \(C(n)_{S_ 0}\) associates a sheaf of \({\mathcal O}_ S/p^ n {\mathcal O}_ S\)-modules, namely the sheaf of (crystalline) extensions of \(G\) by the augmentation ideal of \(S_ 0\) (over \(\mathbb{Z}_ p)\). A main result is that this functor \(M_ S\) is exact. For all \(G\), \(M_ S(G)\) is canonically isomorphic to the dual of \(M_ S(G^ D)\) (where \(G^ D\) denotes the dual group scheme).
The other results of the paper concern the case \(n=1\): then there is a canonical injection of (resp. surjection onto) the sheaf \(\alpha_ G = \operatorname{Hom} (G, \mathbb{G}_ a)\) (resp. the sheaf \(\omega_ G\) of invariant 1-forms of \(G)\) which is an isomorphism if the Verschiebung (resp. the Frobenius) on \(G\) is zero. Unfortunately \(M_ S(G)\) is not locally free in general, but if the Frobenius map can be lifted to \(S\), the author proves that \(M_ S (G)\) is locally free. Much stronger results are obtained if moreover \(S\) is the spectrum of a noetherian complete local ring \(R\) with perfect residue field: Here under an additional technical assumption on \(S\), or if \(M_ S\) is restricted to groups without multiplicative part, \(M_ S\) is an equivalence of categories onto (the appropriate subcategory of) the category of finite locally free \({\mathcal O}_ S/p {\mathcal O}_ S\)-modules endowed with \({\mathcal O}_ S\)-linear maps \(F\) and \(V\) such that \(F \circ V = p = V \circ F\). A corollary to this result is the classification of \(p\)-divisible groups over such rings \(R\).
The starting point and the main technical and notational reference for this article is P. Berthelot, L. Breen and W. Messing, “Théorie de Dieudonné cristalline II,” Lect. Notes Math. 930 (1982; Zbl 0516.14015).

14L15 Group schemes
14G15 Finite ground fields in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
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