# Dieudonné module

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: WnWn+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.

## Dieudonné rings

If k is a field of characteristic p, its ring of Witt vectors consists of sequences (w1,w2,w3,...) of elements of k, and has an endomorphism σ induced by the Frobenius endomorphism of k, so (w1,w2,w3,...)σ = (wp
1
,wp
2
,wp
3
,...). The Dieudonné ring, often denoted by Ek or Dk, is the non-commutative ring over W(k) generated by 2 elements F and V subject to the relations

FV = VF = p
Fw = wσF
wV = Vwσ.

It is a Z-graded ring, where the piece of degree nZ is a 1-dimensional free module over W(k), spanned by Vn if n≤0 and by Fn if n≥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 equivalent to the opposite of the category of finite commutative p-group schemes over k.

## Examples

• If $X$ is the constant group scheme $\mathbb{Z}/p\mathbb{Z}$ over $k$, then its corresponding Dieudonné module $\mathbf{D}(X)$ is $k$ with $F = \mathrm{Frob}_k$ and $V = 0$.
• For the scheme of p-th roots of unity $X = \mu_p$, then its corresponding Dieudonné module is $\mathbf{D}(X) = k$ with $F = 0$ and $V = \mathrm{Frob}_k^{-1}$.
• For $X = \alpha_p$, defined as the kernel of the Frobenius $\mathbb{G}_{a} \to \mathbb{G}_{a}$, the Dieudonné module is $\mathbf{D}(X) = k$ with $F = V = 0$.
• If $X = E[p]$ 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.