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
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
. 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
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
-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 {
-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.