It is often easier to study formal group schemes using the associated Dieudonne module.
A Dieudonné module is defined to be a module over the Dieudonné ring .
If is an affine commutative unitary group scheme over we construct (in Proposition (codir)) a codirected diagram of Witt modules such that the codirected system and its limit become Dieudonné modules.
This construction can be executed similarly in the Ind category so we have Dieudonné modules for formal group schemes and p-divisible groups.
There is a contravariant equivalence between the category of -torsion formal groups and that of finitely generated -modules.
Codirected diagrams of Witt modules
Let be a perfect field of prime characteristic . Let be the functor assigning to a -ring its (p-adic) Dieudonné ring.
Let be the codirected diagram
where denotes the translation. is sometimes called Verschiebung morphism and satisfies and where is the Frobenius morphism.
With the multiplication obtained by the algebra map for and the induced map the above diagram (1) is not a diagram in the category of Dieudonné modules since is not -linear but satisfies hence we have to redefine the multiplication by scalars in by
with this (where we recall that ) we have -linear map since
For an (affine commutative unipotent) group scheme the codirected systems of Witt modules endows the limit of the hom-spaces with the structure of a Dieudonné module:
(see also group scheme for more context concerning this theorem)
Let be a perfect field of prime characteristic . Since is perfect Frobenius is an automorphism.
On the left we have the category of affine commutative unipotent group schemes. On the right we have the category of all D_k-modules of -torsion. The (contravariant) equivalence is given by
If we recall that the Dieudonné ring of is a -graded ring where the degree -part is the -dimensional free module generated by if and by if , we see that morphisms in can be multiplied by powers of reps. powers of by postcomposition.
Since all the operators are are monomorphisms, we get that are all injective and hence we can identify with a submodule of or explicitly we know that . Thus every element of is killed by a power of .
If we introduce a bit of abstraction we can see the beauty of all this. Let be the noncommutative polynomial ring over the Witt vectors on two indeterminates that satisfy the commutation laws , , and . This is called the Dieudonne ring?. We have a canonical way to consider as a left -module. Thus is a contravariant functor from affine unitary group schemes to the category of -modules with torsion. This turns out to be an anti-equivalence of categories.
Michel Demazure, Lectures on -Divisible Groups
Jesse Kass, notes on Dieudonné modules
(expositional) Aaron Mazel-Gee, Dieudonné modules and the classification of formal groups, pdf