Dieudonné module

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 ${\displaystyle F}$ and ${\displaystyle V}$ 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 ${\displaystyle k}$ of positive characteristic ${\displaystyle p}$ can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring

${\displaystyle D=W(k)\{F,V\}/(FV-p)}$,

which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of ${\displaystyle k}$. The endomorphisms ${\displaystyle F}$ and ${\displaystyle V}$ 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 ${\displaystyle k}$ of order a power of ${\displaystyle p}$ and modules over ${\displaystyle D}$ with finite ${\displaystyle W(k)}$-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf ${\displaystyle CW}$ 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 ${\displaystyle V\colon W_{n}\to W_{n+1}}$, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected ${\displaystyle p}$-group schemes correspond to ${\displaystyle D}$-modules for which ${\displaystyle F}$ is nilpotent, and étale group schemes correspond to modules for which ${\displaystyle F}$ 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 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 ${\displaystyle p}$-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's work on the Shimura–Taniyama conjecture.