Contents

# Contents

## Idea

It is often easier to study formal group schemes using the associated Dieudonne module.

## Definition

A Dieudonné module is defined to be a module over the Dieudonné ring $D_k$.

If $G$ is an affine commutative unitary group scheme over $k$ we construct (in Proposition (codir)) a codirected diagram of Witt modules such that the codirected system $M(G)_n:= Hom(G, W_n(k))$ and its limit $M(G):=lim_n Hom(G, W_n(k))$ 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.

## Properties

There is a contravariant equivalence $p.tor.for.Gr\simeq D_k .Mod.fin$ between the category of $p$-torsion formal groups and that of finitely generated $D_k$-modules.

## Examples

### Codirected diagrams of Witt modules

###### Proposition (codir)

Let $k$ be a perfect field of prime characteristic $p$. Let $W_n(-):kRing\to Ring$ be the functor assigning to a $k$-ring its (p-adic) Dieudonné ring.

Let $\underline W$ be the codirected diagram

(1)$W_{1k}\stackrel{V}{\to}W_{2k}\stackrel{V}{\to}W_{3k}\stackrel{V}{\to}\dots$

where $V:\begin{cases}W_{n} (k)\to W_{n+1} (k) \\ (x_0,x_1,\dots,x_{n-1})\mapsto (0,x_0,x_1,\dots, x_{n-1})\end{cases}$ denotes the translation. $V$ is sometimes called Verschiebung morphism and satisfies $VF=id$ and $FV=p$ where $F$ is the Frobenius morphism.

With the multiplication obtained by the algebra map $k\hookrightarrow A$ for $A\in kRing$ and the induced map $W(k)\to W_n(A)\hookrightarrow W(A)$ the above diagram (1) is not a diagram in the category of Dieudonné modules since $V$ is not $W(k)$-linear but satisfies $T(\lambda a)=\lambda^{1/p}T(a)$ hence we have to redefine the multiplication by scalars in $W(k)$ by

$*:\begin{cases} W (k)\times W_{n} (R)\to W_{n} (R) \\ (a,w)\mapsto a^{p^{1-n}} R\cdot w \end{cases}$

with this (where we recall that $W_n:=coker( T^n:W_k\to W_k)$) we have $W(k)$-linear map since

$\displaystyle V(\lambda\star a)=V(\overline{\lambda}^{p^{1-n}}a)=V(F(\overline{\lambda}^{p^{-n}})a)=\overline{\lambda}^{p^{-n}}V(a)=\lambda\star V(a)$

For an (affine commutative unipotent) group scheme the codirected systems of Witt modules endows the limit of the hom-spaces $hom(G,W_n(k))$ with the structure of a Dieudonné module:

###### Theorem

(III.5, $Acu_k\simeq Tor_V D_kMod$)

Let $k$ be a perfect field of prime characteristic $p$. Since $k$ 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 $V$-torsion. The (contravariant) equivalence is given by

$M:\begin{cases} Acu_k&\to& Tor_V D_kMod \\ G&\mapsto&colim_n Acu_k(G,W_{nk}) \end{cases}$

If we recall that the Dieudonné ring of $k$ is a $\mathbb{Z}$-graded ring where the degree $n$-part is the $1$-dimensional free module generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$, we see that morphisms in $colim_n Acu_k(G,W_{nk})$ can be multiplied by powers of $V$ reps. powers of $F$ by postcomposition.

## Properties

Since all the $V$ operators are are monomorphisms, we get that $Hom(G, W_n(k))\to Hom(G, W_{n+1}(k))$ are all injective and hence we can identify $Hom(G, W_n(k))$ with a submodule of $D(G)$ or explicitly we know that $Hom(G, W_n(k))=\{m \in D(G): V^n(m)=0\}$. Thus every element of $D(G)$ is killed by a power of $V$.

If we introduce a bit of abstraction we can see the beauty of all this. Let $D=\Lambda\{F, V\}$ be the noncommutative polynomial ring over the Witt vectors on two indeterminates that satisfy the commutation laws $Fw=w^p F$, $w^p V=Vw$, and $FV=VF=p$. This is called the Dieudonne ring?. We have a canonical way to consider $D(G)$ as a left $D$-module. Thus $G\mapsto D(G)$ is a contravariant functor from affine unitary group schemes to the category of $D$-modules with $V$ torsion. This turns out to be an anti-equivalence of categories.

• Michel Demazure, Lectures on $p$-Divisible Groups

• Jesse Kass, notes on Dieudonné modules

• (expositional) Aaron Mazel-Gee, Dieudonné modules and the classification of formal groups, pdf