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- Q5275225 subject Q7585268.
- Q5275225 abstract "In mathematics, a Dieudonné module introduced by Dieudonné (1954, 1957b), is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D = W(k){F,V}/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Jean Dieudonné and Pierre Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: Wn → Wn+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles's work on the Shimura–Taniyama conjecture.".
- Q5275225 wikiPageExternalLink cartier1.pdf.
- Q5275225 wikiPageWikiLink Q1082992.
- Q5275225 wikiPageWikiLink Q1756171.
- Q5275225 wikiPageWikiLink Q184433.
- Q5275225 wikiPageWikiLink Q18848.
- Q5275225 wikiPageWikiLink Q190109.
- Q5275225 wikiPageWikiLink Q2584927.
- Q5275225 wikiPageWikiLink Q371957.
- Q5275225 wikiPageWikiLink Q574633.
- Q5275225 wikiPageWikiLink Q649469.
- Q5275225 wikiPageWikiLink Q703577.
- Q5275225 wikiPageWikiLink Q7585268.
- Q5275225 wikiPageWikiLink Q7644139.
- Q5275225 comment "In mathematics, a Dieudonné module introduced by Dieudonné (1954, 1957b), is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators.".
- Q5275225 label "Dieudonné module".