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In mathematics, a Dieudonné module introduced by , is a module over the non-commutative Dieudonné ring, which is generated over the ring of
Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field o ...
s by two special endomorphisms F and V called the
Frobenius Frobenius is a surname. Notable people with the surname include: * Ferdinand Georg Frobenius (1849–1917), mathematician ** Frobenius algebra ** Frobenius endomorphism ** Frobenius inner product ** Frobenius norm ** Frobenius method ** Frobenius g ...
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)\/(FV-p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. The endomorphisms F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. 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 In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
), since it is constructed by taking a
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of finite length Witt vectors under successive Verschiebung maps V\colon W_n \to W_, 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. Tadao Oda's 1967 thesis gave a connection between Dieudonné modules and the first
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
of
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
, and at about the same time,
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
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
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...
's work on the Shimura–Taniyama conjecture.


Dieudonné rings

If k is a perfect field of characteristic p, its ring of
Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field o ...
s consists of sequences (w_1,w_2,w_3,\dots) of elements of k, and has an endomorphism \sigma induced by the
Frobenius endomorphism In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime number, prime characteristic (algebra), ...
of k, so (w_1,w_2,w_3,\dots)^\sigma=(w^p_1,w^p_2,w^p_3,\dots). The Dieudonné ring, often denoted by E_k or D_k, is the non-commutative ring over W(k) generated by 2 elements F and V subject to the relations :FV = VF = p :Fw = w^\sigma F :wV = Vw^\sigma. It is a \mathbb-graded ring, where the piece of degree is a 1-dimensional
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
over W(k), spanned by V^ if n\leq 0 and by F^n if n\geq 0. Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by F and V.


Dieudonné modules and groups

Special sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finite length modules over the Dieudonné ring form an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category o ...
equivalent to the opposite of the category of finite commutative p-group schemes over k.


Examples

* If X is the constant group scheme \mathbb/p\mathbb over k, then its corresponding Dieudonné module \mathbf(X) is k with F = \mathrm_k and V = 0 . * For the scheme of p-th roots of unity X = \mu_p, then its corresponding Dieudonné module is \mathbf(X) = k with F = 0 and V = \mathrm_k^. * For X = \alpha_p, defined as the kernel of the Frobenius \mathbb_ \to \mathbb_, the Dieudonné module is \mathbf(X) = k with F = V = 0 . * If X = E is the p-torsion of an elliptic curve over k (with p-torsion in k), then the Dieudonné module depends on whether E is supersingular or not.


Dieudonné–Manin classification theorem

The Dieudonné–Manin classification theorem was proved by and . It describes the structure of Dieudonné modules over an algebraically closed field k up to "isogeny". More precisely, it classifies the finitely generated modules over D_k /p/math>, where D_k is the Dieudonné ring. The category of such modules is semisimple, so every module is a direct sum of simple modules. The simple modules are the modules E_ where r and s are coprime integers with r>0. The module E_ has a basis over W(k) /p/math> of the form v, Fv, F^2 v,\dots, F^v for some element v, and F^r v=p^s v. The rational number s/r is called the slope of the module.


The Dieudonné module of a group scheme

If G is a commutative group scheme, its Dieudonné module D(G) is defined to be \text(G,W), defined as \lim_n\text(G,W_n) where W is the formal Witt group scheme and W_n is the truncated Witt group scheme of Witt vectors of length n. The Dieudonné module gives antiequivalences between various sorts of commutative group schemes and left modules over the Dieudonné ring D. *Finite commutative group schemes of p-power order correspond to D modules that have finite length over W. *Unipotent affine commutative group schemes correspond to D modules that are V-torsion. *p-divisible groups correspond to D-modules that are finitely generated free W-modules, at least over perfect fields.


Dieudonné crystal

A Dieudonné crystal is a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
D together with homomorphisms F:D^p\to D and V:D\to D^p satisfying the relations VF=p (on D^p), FV=p (on D). Dieudonné crystals were introduced by . They play the same role for classifying algebraic groups over schemes that Dieudonné modules play for classifying algebraic groups over fields.


References

* * * * * *. *


External links

* {{DEFAULTSORT:Dieudonne module Algebraic groups