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\begin{document}

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\section*{Demazure, lectures on p-divisible groups, III.6, Dieudonné modules (p-torsion finite k-groups)}

[[!redirects III.6, Dieudonné modules (p-torsion finite k-groups)]]

This entry is about a section of the text

\begin{itemize}%
\item Michel [[Demazure, lectures on p-divisible groups]], \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}

\end{itemize}
Recall from [[relation of certain classes of k-groups]] the following:

\begin{enumerate}%
\item $Feu_k$ denotes the category of formal \'e{}tale unipotent affine $k$-groups.


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


\item $W(k)$ denotes the Witt ring over $k$.


\item For $D_k$ see [[D\_k-module]] in [[III.5, Dieudonné modules (affine unipotent groups) ]].


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


\end{enumerate}
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 $F$).

\begin{prop}
\label{}\hypertarget{}{}
(formulation of the statement is unclear) The functor

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

\begin{enumerate}%
\item $Feu_k\to Tor_V D_k-Mod\hookrightarrow M(G)$ between the category of affine \'e{}tale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $V$ on which $F$ is bijective.


\item $Fiu_k\to Tor_F D_k-Mod\hookrightarrow M(G)$ between the category of affine \'e{}tale 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.



\end{enumerate}
(\hyperlink{lecturesOnPDivGroups}{Demazure p.69})

\end{prop}
\begin{proof}
This follows from the theorem, and the fact that if $G$ is finite, then G is \'e{}tale (resp, infinitesimal) if and only if $F_G$ is an isomorphism (resp. $F_G^n = 0$ for large $n$).

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item If $G=(\mathbb{Z}/p\mathbb{Z})_k\in Feu_k$, then $M(G)=k$ with $F=1$, $V=0$.


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



\end{enumerate}
\end{example}
\begin{cor}
\label{}\hypertarget{}{}
For $G\in Feu_k$ or $G\in Fiu_k$, we have

\begin{displaymath}
rk(G)= p^{length(M(G))
\end{displaymath}
\end{cor}
\begin{prop}
\label{}\hypertarget{}{}
Let $m,n$ be two positive integers. Then

\begin{enumerate}%
\item 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

\end{enumerate}
\begin{displaymath}
\lambda_{n,m}: D_k/(D_k F^m + D V^n)\to M(m^W_n)
\end{displaymath}
which is bijective.

\end{prop}
\begin{theorem}
\label{}\hypertarget{}{}
There is an isomorphism

\begin{displaymath}
M(D(G))\to M(G)^*
\end{displaymath}
In prose this means that the autoduality $G\mapsto D(G)$ of $Fiu_k$ corresponds via the Dieudonn\'e{}-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 $F$.

\end{theorem}
\begin{defn}
\label{}\hypertarget{}{}
(Dieudonn\'e{}-module of an infinitesimal multiplicative $k$-group)

Let $G\in Fim_k$. Then the \emph{Dieudonn\'e{}-module of $M(G)*$ is defined by}

\begin{displaymath}
M(G)=M(D(G))^*
\end{displaymath}
It follows by the [[Cartier duality]] between $Fim_k$ and $Feu_k$ that the functor $G\mapsto M(G)$ induces a contravariant equivalence

\begin{displaymath}
Fim_k\to  fin Tor_F Bij_V D_k-Mod
\end{displaymath}
between $Fim_k$ and the category of all $D_k$-modules of finite length on which $F$ is nilpotent and $V$ is bijective.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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

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

\begin{displaymath}
F(\lambda \otimes \chi)=\lambda^{(p)}\otimes p\chi
\end{displaymath}
\begin{displaymath}
V(\lambda \otimes \chi)=\lambda^{(p^{-1})}\otimes p\chi
\end{displaymath}
\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
a) The Dieudonn\'e{} functor

\begin{displaymath}
\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}
\end{displaymath}
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

\begin{displaymath}
F_M(\lambda m)=\lambda^{(p)} F_M(m)
\end{displaymath}
\begin{displaymath}
V_M(\lambda^{(p)}m)=\lambda V_M(m)
\end{displaymath}
\begin{displaymath}
F_M V_M=V_M F_M=p\cdot id_M
\end{displaymath}
b) $G$ is \'e{}tale, 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

\begin{displaymath}
M(D(G))=M(G)^*
\end{displaymath}
\end{theorem}
\hypertarget{references}{}\subsection*{{References}}\label{references}

 Michel Demazure, lectures on p-divisible groups \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}



\end{document}