nLab
Dieudonne module

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 $M(G{)}_n:=\mathrm{Hom}(G,W_n(k))$ and its limit $M(G):={\lim }_n\mathrm{Hom}(G,W_n(k))$ 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 $p.\mathrm{tor}.\mathrm{for}.\mathrm{Gr}\simeq D_k.\mathrm{Mod}.\mathrm{fin}$ 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 $W_n(-):\mathrm{kRing}\to \mathrm{Ring}$ be the functor assigning to a $k$-ring its (p-adic) Dieudonné ring.

Let $\underline{W}$ be the codirected diagram

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

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

With the multiplication obtained by the algebra map $k\hookrightarrow A$ for $A\in \mathrm{kRing}$ 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

$$*:\{\begin{array}{l}W(k)\times W_n(R)\to W_n(R)\\ (a,w)\mapsto a^{p^{1-n}}R\cdot w\end{array}$$*:\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 $W_n:=\mathrm{coker}(T^n:W_k\to W_k)$) we have $W(k)$-linear map since

$${\displaystyle V(\lambda \star a)=V({\overline{\lambda }}^{p^{1-n}}a)=V(F({\overline{\lambda }}^{p^{-n}})a)={\overline{\lambda }}^{p^{-n}}V(a)=\lambda \star V(a)}$$\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 $\mathrm{hom}(G,W_n(k))$ with the structure of a Dieudonné module:

Theorem

(III.5, ${\mathrm{Acu}}_k\simeq {\mathrm{Tor}}_V{D}_{\mathrm{kMod}}$)

(see also group scheme for more context concerning this theorem)

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

$$M:\{\begin{array}{lll}{\mathrm{Acu}}_k& \to & {\mathrm{Tor}}_V{D}_{\mathrm{kMod}}\\ G& \mapsto & {\mathrm{colim}}_n{\mathrm{Acu}}_k(G,W_{\mathrm{nk}})\end{array}$$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 $n<0$ and by $F^n$ if $n>0$, we see that morphisms in ${\mathrm{colim}}_n{\mathrm{Acu}}_k(G,W_{\mathrm{nk}})$ 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 $\mathrm{Hom}(G,W_n(k))\to \mathrm{Hom}(G,W_{n+1}(k))$ are all injective and hence we can identify $\mathrm{Hom}(G,W_n(k))$ with a submodule of $D(G)$ or explicitly we know that $\mathrm{Hom}(G,W_n(k))=\{m\in D(G):V^n(m)=0\}$. 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 $\mathrm{Fw}=w^{p}F$, $w^{p}V=\mathrm{Vw}$, and $\mathrm{FV}=\mathrm{VF}=p$. 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.

References

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

• Jesse Kass, notes on Dieudonné modules