# nLab Demazure, lectures on p-divisible groups, III.6, Dieudonné modules (p-torsion finite k-groups)

This entry is about a section of the text

Recall from relation of certain classes of k-groups? the following:

1. $Feu_k$ denotes the category of formal étale unipotent affine $k$-groups.

2. $Fiu_k$ denotes the category of formal infinitesimal unipotent $k$-groups.

3. $W(k)$ denotes the Witt ring over $k$.

4. For $D_k$ see D_k-module? in III.5, Dieudonné modules (affine unipotent groups).

5. $M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}$

is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.

Recall moreover from III.5, Dieudonné modules (affine unipotent groups) that $Tor_V D_k-Mod:=Acu_k(G,W_{nk})=\{m\in M(G)|V^n m =0\}$ is a submodule.

which are $W(k)$-modules of finite length, killed by a power of $V$, Definition Verschiebung morphism, and on which $F$, Definition Frobenius morphism, is bijective (resp. and killed by a power of $F$).

###### Proposition

(formulation of the statement is unclear) The functor

$M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}$

which is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules induces the following contravariant equivalences of categories:

1. $Feu_k\to Tor_V D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $V$ on which $F$ is bijective.

2. $Fiu_k\to Tor_F D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $F$ (and killed by a power of $V$ ?)on which $F$ is bijective.

###### Proof

This follows from the theorem, and the fact that if $G$ is finite, then G is étale (resp, infinitesimal) if and only if $F_G$ is an isomorphism (resp. $F_G^n = 0$ for large $n$).

###### Example
1. If $G=(\mathbb{Z}/p\mathbb{Z})_k\in Feu_k$, then $M(G)=k$ with $F=1$, $V=0$.

2. If $G=p \alpha_k\in Fiu_k$, then $M(G)=k$ with $F=0$, $V=0$.

###### Corollary

For $G\in Feu_k$ or $G\in Fiu_k$, we have

rk(G)= p^{length(M(G))
###### Proposition

Let $m,n$ be two positive integers. Then

1. The canonical injection $m^{W_n}\to W_n$ defines an element $u\in M(m^{W_n})$ satisfying $V^n u= F^n= 0$. This gives a map
$\lambda_{n,m}: D_k/(D_k F^m + D V^n)\to M(m^W_n)$

which is bijective.

###### Theorem

There is an isomorphism

$M(D(G))\to M(G)^*$

In prose this means that the autoduality $G\mapsto D(G)$ of $Fiu_k$ corresponds via the Dieudonné-functor $M$ to the autoduality $M\to M^*$ in the category $fin Tor_{V,F}D_k-Mod$ of $D_k$-modules of finite length killed by a power of $V$ and $F$.

###### Definition

(Dieudonné-module of an infinitesimal multiplicative $k$-group)

Let $G\in Fim_k$. Then the Dieudonné-module of $M(G)*$ is defined by

$M(G)=M(D(G))^*$

It follows by the Cartier duality between $Fim_k$ and $Feu_k$ that the functor $G\mapsto M(G)$ induces a contravariant equivalence

$Fim_k\to fin Tor_F Bij_V D_k-Mod$

between $Fim_k$ and the category of all $D_k$-modules of finite length on which $F$ is nilpotent and $V$ is bijective.

###### Remark

Let $G\in Fimd_k$ (i.e. $G\in Fim_k$ and $G$ diagonalizable).

$G=D(\Gamma_k)$. Then $D(G)\simeq \Gamma_k$, and

$M(D(G))=colim Acu_k(\Gamma_k,W_{nk})=colimGr(\Gamma,W_n(k))=Gr(\Gamma,W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z},W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z} \Gamma, W_\infty)$

and hence

$M(G)\simeq(W(\overline k)\otimes_\mathbb{Z}\Gamma)^\pi$

where $W_\infty=Quot(W(k))/W(k)=colim_n W_n(k)=\underline W(k)$, see p.66.

For $F$ and $V$ we have

$F(\lambda \otimes \chi)=\lambda^{(p)}\otimes p\chi$
$V(\lambda \otimes \chi)=\lambda^{(p^{-1})}\otimes p\chi$
###### Theorem

a) The Dieudonné functor

$\begin{cases} F_p_k=Fiu_k\times Feu_k\times Fim_k\to (fin W(k)-Mod,F,V) \\ G\to M(G) \end{cases}$

is a contravariant equivalence between all finite $k$-groups of $p$-torsion, and the category of all triples $(M,F_M,V_M)$ where $M$ is a finite length $W(k)$-module and $F_M$, $V_M$ two group endomorphisms of $M$ satisfying

$F_M(\lambda m)=\lambda^{(p)} F_M(m)$
$V_M(\lambda^{(p)}m)=\lambda V_M(m)$
$F_M V_M=V_M F_M=p\cdot id_M$

b) $G$ is étale, infinitesimal, unipotent or multiplicative according as $F_M$ is isomorphic, $F_M$ is nilpotent, $V_M$ is nilpotent, or $V_M$ is isomorphic

c) For any $G\in Fp_k$ we have $rk(G)=p^{length M(G)}$.

d) If $k$ is a perfect extension of $k$, there exists a functorial isomorphism

$M(D(G))=M(G)^*$

Michel Demazure, lectures on p-divisible groups web