# nLab Demazure, lectures on p-divisible groups, III.8, Dieudonné modules (p-divisible groups)

This entry is about a section of the text

###### Lemma

Let $M_{n+1}\stackrel{\pi_n}{\to}M_n\stackrel{\pi_{n-1}}{\to}\cdots\stackrel{\pi_n}{\to}M_1$ a system of $W(k)$-modules? such that for all $n$

1. $M_{n+1}\stackrel{p^n}{\to}M_{n+1}\stackrel{\pi_n}{\to}M_n\to 0$ is exact

2. $M_n$ is of finite length.

then $M:=lim M_n$ is a finitely generated $W(k)$-module and the canonical map $M\to M_n$ identifies $M_n\simeq M/p^n M$

###### Definition

($p$-torsion formal group) A formal group $G$ is called $p$-torsion formal group if

1. $G=\cup ker p^n id_G$

2. $ker p id_G$ is finite.

There are exact sequences

$0\to ker p^n \to ker p^{n+1}\stackrel{p^n}{\to}ker p^{n+1}$
$0\to ker p^n \to ker p^{n+m}\stackrel{p^n}{\to}ker p^m$

showing by induction the also $ker p^n$ is finite for all $n$. Define $M(G)= colim M(ker p^n)$

###### Theorem

$G\to M(G)$ is a (contravariant) equivalence between the category of $p$-torsion?formal groups and the category of tuples $(M,F_M,V_M)$ where M is a finitely generated $W(k)$-module and $F_M$, $V_M$ to groups of endomorphisms of $M$ with

$F_M(wm)=w^{(p)}F_M (m)$
$V_M(w^{(p)}_m)=w V_n(m)$
$F_M V_M=V_M F_M=p id_M$

It follows from the lemma that $M(G)$ is finitely generated and that

$M_n=M(G)/p^n M(G)$

Conversely if $M$ is as before we define $G:=colim G_n$ where $M(G_n)=M/p^n M$

Moreover we have:

1. $G$ is finite iff $M(G)$ is finite and in that case $M(G)$ is the same as in § 7.

2. $G$ is $p$-divisible iff $M(G)$ is torsion-less (= free) and $height(G)=dim M(G)$.

3. For any perfect extension? $K/k$ there is a functorial isomorphism $M(G\otimes_k K)\simeq W(k)\otimes_{W(k)}M(G)$

4. If $G$ is $p$-divisible with Serre dual? $G^\prime$ then $M(G^\prime)=Mod_{W(k)}(M(G),W(k)$ with

$(F_{M(G^\prime)}f)(m)=f(V_M m)(p)$

and

$(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{-1})}$
###### Theorem

a) The Dieudonné functor

$\begin{cases} Torf_p\to (fin W(k)-Mod,F,V) \\ G\to M(G) \end{cases}$

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

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

It follows from the lemma that $M(G)$ is finitely generated and $M_n\simeq M(G)/p^n M(G)$. Conversely if $M$ is as before, then we define $G$ as $colim G_n$ where $M(G_n)= M/ p^n M$.

###### Remark

From the definition and what we already verified follows:

1. $G$ is finite iff $M(G)$ is finite, and in that case $M(G)$ is the same as defined in § 7.

2. $G$ is finite iff $M(G)$ is torsion-less (= free), and $height(G)=dim M(G)$.

3. For any perfect extension $K/k$, there is a functorial isomorphism $M(G\otimes_k K)\simeq W(K)\otimes_{W(k)} M(G)$.

4. If $G$ is $p$-divisible, with Serre dual $G^\prime$, then $M(G^\prime)=Mod_{W(k)}M(G)$, with $F_{M(G^\prime)} f)(m)=f(V_M m)^{(p)}$ and $(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{(-1)}}$.

Michel Demazure, lectures on p-divisible groups web