This entry is about a section of the text
Recall from relation of certain classes of k-groups? the following:
denotes the category of formal étale unipotent affine -groups.
denotes the category of formal infinitesimal unipotent -groups.
denotes the Witt ring over .
For see D_k-module? in III.5, Dieudonné modules (affine unipotent groups).
is a contravariant functor from affine commutative unipotent -groups to the category of -modules.
Recall moreover from III.5, Dieudonné modules (affine unipotent groups) that is a submodule.
(formulation of the statement is unclear) The functor
which is a contravariant functor from affine commutative unipotent -groups to the category of -modules induces the following contravariant equivalences of categories:
between the category of affine étale unipotent -groups to the category of -modules of finite length, killed by a power of on which is bijective.
between the category of affine étale unipotent -groups to the category of -modules of finite length, killed by a power of (and killed by a power of ?)on which is bijective.
This follows from the theorem, and the fact that if is finite, then G is étale (resp, infinitesimal) if and only if is an isomorphism (resp. for large ).
If , then with , .
If , then with , .
For or , we have
Let be two positive integers. Then
which is bijective.
There is an isomorphism
In prose this means that the autoduality of corresponds via the Dieudonné-functor to the autoduality in the category of -modules of finite length killed by a power of and .
(Dieudonné-module of an infinitesimal multiplicative -group)
Let . Then the Dieudonné-module of is defined by
It follows by the Cartier duality between and that the functor induces a contravariant equivalence
between and the category of all -modules of finite length on which is nilpotent and is bijective.
Let (i.e. and diagonalizable).
. Then , and
where , see p.66.
For and we have
a) The Dieudonné functor
is a contravariant equivalence between all finite -groups of -torsion, and the category of all triples where is a finite length -module and , two group endomorphisms of satisfying
b) is étale, infinitesimal, unipotent or multiplicative according as is isomorphic, is nilpotent, is nilpotent, or is isomorphic
c) For any we have .
d) If is a perfect extension of , there exists a functorial isomorphism
Michel Demazure, lectures on p-divisible groups web