Finite locally free group schemes in characteristic \(p\) and Dieudonné modules
type: Item
instance of: scholarly article
title: Finite locally free group schemes in characteristic \(p\) and Dieudonné modules
zbMATH Open document ID: 0812.14030
DOI: 10.1007/BF01232664
published in: Inventiones Mathematicae
publication date: 1995-05-23T00:00:00Z
full work available at URL: https://eudml.org/doc/144143
review text: 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 \(\alpha_ G = \Hom (G, \mathbb{G}_ a)\) (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…
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 549643
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: \(p\)-group schemes
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: \(p\)- divisible groups
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: Frobenius map
author: Aise Johan de Jong
reviewed by: Frank Herrlich
MaRDI profile type: Publication
cites work: Théorie de Dieudonne cristalline. II
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cites work: Universal extensions and one dimensional crystalline cohomology
cites work: The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes
cites work: Commutative group schemes
cites work: Group schemes of prime order
description: scientific article; zbMATH DE number 549643