A module over the non-commutative Dieudonné ring
In mathematics, a Dieudonné module introduced by Jean 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 formula_1 and formula_2 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 formula_3 of positive characteristic formula_4 can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring 
formula_5, 
which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of formula_3. The endomorphisms formula_1 and formula_2 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 formula_3 of order a power of formula_4 and modules over formula_11 with finite formula_12-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf formula_13 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 formula_14, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected formula_4-group schemes correspond to formula_11-modules for which formula_1 is nilpotent, and étale group schemes correspond to modules for which formula_1 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 of abelian varieties, and at about the same time, Alexander Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze formula_4-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's work on the Shimura–Taniyama conjecture.
Dieudonné rings.
If formula_3 is a perfect field of characteristic formula_4, its ring of Witt vectors consists of sequences formula_22 of elements of formula_3, and has an endomorphism formula_24 induced by the Frobenius endomorphism of formula_3, so formula_26. The Dieudonné ring, often denoted by formula_27 or formula_28, is the non-commutative ring over formula_12 generated by 2 elements formula_1 and formula_2 subject to the relations
formula_32
formula_33
formula_34.
It is a formula_35-graded ring, where the piece of degree formula_36 is a 1-dimensional free module over formula_12, spanned by formula_38 if formula_39 and by formula_40 if formula_41.
Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by formula_1 and formula_2.
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 equivalent to the opposite of the category of finite commutative formula_4-group schemes over formula_3.
Dieudonné–Manin classification theorem.
The Dieudonné–Manin classification theorem was proved by Dieudonné (1955) and Yuri Manin (1963). It describes the structure of Dieudonné modules over an algebraically closed field formula_3 up to "isogeny". More precisely, it classifies the finitely generated modules over formula_69, where formula_28 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 formula_71 where formula_72 and formula_73 are coprime integers with formula_74. The module formula_71 has a basis over formula_76 of the form formula_77 for some element formula_78, and formula_79. The rational number formula_80 is called the slope of the module.
The Dieudonné module of a group scheme.
If formula_81 is a commutative group scheme, its Dieudonné module formula_82 is defined to be formula_83, defined as formula_84 where formula_85 is the formal Witt group scheme and 
formula_86 is the truncated Witt group scheme of Witt vectors of length formula_87. 
The Dieudonné module gives antiequivalences between various sorts of commutative group schemes and left modules over the Dieudonné ring formula_11.
Dieudonné crystal.
A Dieudonné crystal is a crystal formula_11 together with homomorphisms formula_98 and formula_99 satisfying the relations formula_100 (on formula_101), formula_102 (on formula_11). 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.