This is a note on the construction of Dieudonne modules (over a perfect field), prepared for a student seminar on
-adic Hodge theory at Harvard, Fall 2012. Our main references are [1] and [2].
Dieudonne modules
Let
be a perfect field of characteristic
. Our goal is to classify finite (commutative)
-groups of
-power order using (semi-)linear algebraic data. Recall every such
-group can be decomposed as
Let
be the Frobenius and
be the Verschiebung, then
- on
:
is an isomorphism and
is nilpotent.
- on
:
is nilpotent and
is nilpotent.
- on
:
is nilpotent and
is an isomorphism.
The constant group

is of etale-local type,

acts as the identity and

acts as 0. Dually, the multiplicative group

is of local-etale type,

acts as 0 and

acts as the identity. The self-dual group

is of local-local type and both

and

acts as 0.
The etale part is relatively easy to deal with: it is same as giving the data of a finite
-module. Over
, the etale part is a constant group of
-power order and can be determined by its Pontryagin dual
. Of course, the local part cannot be detected at the level of points: we would like some enhancement of
to also detect the local part. A natural candidate is using the Witt ring scheme
. Let
be the ring scheme of Witt vectors of length
. We have at the level of
-points
and in general,
Motivated by this, we introduce the local group
to be the kernel of
on
. These are the replacements of
for local groups.
Staring at the action of
and
on
motivates the following definition.
Let

be the automorphism lifting

on

. Let
![$D_k=W(k)[F,V]$](./latex/latex2png-Dieudonne_149503115_-5.gif)
be the
Dieudonne ring (noncommutative unless

) subject to the relations

,

and

.
Let

be of local-local type. We define the (contravariant)
Dieudonne module of

to be

Notice

becomes a left

-module via the action of

on

.
The functor

gives an exact anti-equivalence of categories between {finite

-group schemes of local-local type} and {left

-modules of finite

-length with

,

nilpotent}.
We will not prove this theorem in detail but let us explain why the functor lies in the desired target.
The first part follows from the functoriality of

and

. The second part follows from induction and the fact that

has

-length 1. Notice that

is always injective. To show the surjectivity, one uses the fact that if

and

on

, then any homomorphism

for

and

factors uniquely through

(again by functoriality of

and

).
¡õ
For a general finite
-group, we define its Dieudonne module for the three parts in its decomposition separately.
Suppose

is etale-local, then we define

. Suppose

is local-etale, then we define

, where the dual outside is given by
![$M^\vee=\Hom_{W(k)}(M, W(k)[1/p]/W(k))$](./latex/latex2png-Dieudonne_224878624_-6.gif)
. In general, we decompose

and define its (contravariant)
Dieudonne module
Let us also mention an important property that recovers the cotangent space of
from its Dieudonne module.
There is an natural isomorphism of vector spaces

.
Notice that the tangent space
![$$T_{G,0}:=\ker (G(k[\varepsilon])\rightarrow G(k))\cong \Hom(G^\vee, \mathbb{G}_a)=\Hom(G^\vee,W_1).$$](./latex/latex2png-Dieudonne_21493773_.gif)
Since

on any

, we know that

as desired.
¡õ
We summarize the main theorem of Dieudonne theory as follows.
Fontaine's uniform construction
One unsatisfying aspect of the above construction of the Dieudonne functor is that definition for the local-etale (multiplicative) part seems a bit artificial. In the second part of this talk, we shall describe a uniform construction due to Fontaine.
Suppose

is a

-algebra. By shifting to the left, we see that every element of

is represented by a
covector 
,

. Let

be the

-th universal addition polynomial for Witt vectors, then the addition rule on covectors

is given by

for

(which stabilizes). We denote the

-group scheme obtained this way by

, called the
group of unipotent Witt covectors. This is simply a reformulation of what we used to detect the unipotent part of finite

-group schemes of power order.
In order to also detect the multiplicative part, Fontaine generalizes it to the following.
Suppose

is a

-algebra. We define

to consist of

,

for which there exists

such that the ideal generated by

is nilpotent. Then

and indeed the addition rule also extends to

. We call the

-group scheme

the
group of Witt covectors. In particular,

.
The functor

from finite

-algebras to groups

is pro-represented by a formal

-group

.
Now we can define the Dieudonne module for the larger category of formal
-groups over
using
.
For any formal

-group

over

, we define its (contravariant)
Dieudonne module 
.
For any
![$W(k)[F]$](./latex/latex2png-Dieudonne_264375767_-5.gif)
-profinite topological

-module

, we define

for any finite

-algebra

. Then

is a formal

-group over

(as it is left exact).
(Fontaine)
The functor

gives an exact anti-equivalence of categories between {formal

-group schemes over

} and {
![$W(k)[F]$](./latex/latex2png-Dieudonne_264375767_-5.gif)
-profinite topological

-modules}.

is its quasi-inverse.
References
[1]Richard Pink, Finite group schemes, 2004, www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf .
[2]Fontaine, J.M., Groupes p-divisibles sur les corps locaux, Société mathématique de France, 1977.
[3]Oda, T., The first de Rham cohomology group and Dieudonné modules, Ann. Sci. Ecole Norm. Sup.(4) 2 (1969), no.1, 63--135.