Dieudonne module



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


A Dieudonné module is defined to be a module over the Dieudonné ring D kD_k.

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


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


Codirected diagrams of Witt modules

Proposition (codir)

Let kk be a perfect field of prime characteristic pp. Let W n():kRingRingW_n(-):kRing\to Ring be the functor assigning to a kk-ring its (p-adic) Dieudonné ring.

Let W̲\underline W be the codirected diagram

(1)W 1kVW 2kVW 3kVW_{1k}\stackrel{V}{\to}W_{2k}\stackrel{V}{\to}W_{3k}\stackrel{V}{\to}\dots

where V:{W n(k)W n+1(k) (x 0,x 1,,x n1)(0,x 0,x 1,,x n1)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. VV is sometimes called Verschiebung morphism? and satisfies VF=idVF=id and FV=pFV=p where FF is the Frobenius morphism.

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

*:{W(k)×W n(R)W n(R) (a,w)a p 1nRw*:\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 kW k)W_n:=coker( T^n:W_k\to W_k)) we have W(k)W(k)-linear map since

V(λa)=V(λ¯ p 1na)=V(F(λ¯ p n)a)=λ¯ p nV(a)=λV(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 hom(G,W n(k))hom(G,W_n(k)) with the structure of a Dieudonné module:


(III.5, Acu kTor VD kModAcu_k\simeq Tor_V D_kMod)

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

Let kk be a perfect field of prime characteristic pp. Since kk 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 VV-torsion. The (contravariant) equivalence is given by

M:{Acu k Tor VD kMod G colim nAcu k(G,W nk)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 kk is a \mathbb{Z}-graded ring where the degree nn-part is the 11-dimensional free module generated by V nV^{-n} if n<0n\lt 0 and by F nF^n if n>0n\gt 0, we see that morphisms in colim nAcu k(G,W nk)colim_n Acu_k(G,W_{nk}) can be multiplied by powers of VV reps. powers of FF by postcomposition.


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

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


  • Michel Demazure, Lectures on pp-Divisible Groups

  • Jesse Kass, notes on Dieudonné modules

Revised on July 21, 2012 22:28:56 by Stephan Alexander Spahn (