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

1. on : is an isomorphism and is nilpotent.
2. on : is nilpotent and is nilpotent.
3. on : is nilpotent and is an isomorphism.
Example 1 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.

Remark 1 Recall that the Frobenius and Verschiebung acts on as Then and . Moreover, we have exact sequences It follows that is a successive extension of copies of .

Staring at the action of and on motivates the following definition.

Definition 1 Let be the automorphism lifting on . Let be the Dieudonne ring (noncommutative unless ) subject to the relations , and .
Definition 2 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 .
Remark 2 The action of on needs to be modified so that it is compatible with the transition maps between the 's. Namely, the action of on is modified to be multiplication by .
Theorem 1 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.

Proposition 1 Suppose is of local-local type.
1. If and on , then and annihilate .
2. .
Proof 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
Remark 3 One can further identify . The injectivity is clear since acts non-trivially on . For the surjectivity, one computes the length of as a module to be , which is equal to .

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

Definition 3 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
Remark 4 The formation of Dieudonne module commutes with taking the dual, i.e., . This reduces to the local-local case, which essentially reduces to the duality between and .

Let us also mention an important property that recovers the cotangent space of from its Dieudonne module.

Proposition 2 There is an natural isomorphism of vector spaces .
Proof Notice that the tangent space Since on any , we know that as desired. ¡õ
Remark 5 It is amusing to compare it with the exact sequence (in any characteristic) associated to an abelian variety : Oda [3] proved that is canonically isomorphic to over any perfect field of characteristic and the Hodge filtration can be identified with .

We summarize the main theorem of Dieudonne theory as follows.

Theorem 2 There is an exact additive anti-equivalence of categories between {finite -group schemes of -power order} and {left -modules of finite -length}. satisfying:
1. .
2. .
3. is canonically isomorphic to the cotangent space of .
4. (extension of scalars) for any extension of perfect fields, .
Example 2 Suppose is of order . Then is a -module of rank 1 and acts as 0 on . Hence is a one dimensional -vector space with the action of and .
1. For , acts as and acts as 0.
2. For , acts as 0 and acts as .
3. For , both and acts as 0.
Remark 6 Taking limit in the main theorem gives us an anti-equivalence between -divisible groups over (of height ) and left -modules which are free as -modules (of rank ).

## 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.

Definition 4 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.

Definition 5 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, .
Remark 7 We endow with the discrete topology and with the induced subspace topology from the product topology. Then becomes a complete and separated topological group. Endow with the -adic topology from , then acts on continuously and makes a topological -module, which is torsion, complete and separated.
Theorem 3 The functor from finite -algebras to groups is pro-represented by a formal -group .
Remark 8 More explicitly, , where is the profinite completion of

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

Definition 6 For any formal -group over , we define its (contravariant) Dieudonne module .
Remark 9 Suppose is the affine algebra of , then is closed under the topology defined above. So is also a topological -module. Moreover, one can check it is -profinite, i.e., as a -module, the quotient by any of its open submodules has finite length.
Definition 7 For any -profinite topological -module , we define for any finite -algebra . Then is a formal -group over (as it is left exact).
Theorem 4 (Fontaine) The functor gives an exact anti-equivalence of categories between {formal -group schemes over } and {-profinite topological -modules}. is its quasi-inverse.
Remark 10 When restricting to the subcategory of finite -groups of -power order and the subcategory of -divisible groups over , we recover the classical Dieudonne functor with all the desired properties.

#### 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.