This entry is about a section of the text
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 $n$
$M_{n+1}\stackrel{p^n}{\to}M_{n+1}\stackrel{\pi_n}{\to}M_n\to 0$ is exact
$M_n$ 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 $M_n\simeq M/p^n M$
($p$-torsion formal group) A formal group $G$ is called $p$-torsion formal group if
$G=\cup ker p^n id_G$
$ker p id_G$ is finite.
There are exact sequences
showing by induction the also $ker p^n$ is finite for all $n$. Define $M(G)= colim M(ker p^n)$
$G\to M(G)$ 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
It follows from the lemma that $M(G)$ is finitely generated and that
Conversely if $M$ is as before we define $G:=colim G_n$ where $M(G_n)=M/p^n M$
Moreover we have:
$G$ is finite iff $M(G)$ is finite and in that case $M(G)$ is the same as in § 7.
$G$ is $p$-divisible iff $M(G)$ is torsion-less (= free) and $height(G)=dim M(G)$.
For any perfect extension? $K/k$ there is a functorial isomorphism $M(G\otimes_k K)\simeq W(k)\otimes_{W(k)}M(G)$
If $G$ is $p$-divisible with Serre dual? $G^\prime$ then $M(G^\prime)=Mod_{W(k)}(M(G),W(k)$ with
and
a) The Dieudonné functor
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
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 $M(G_n)= M/ p^n M$.
From the definition and what we already verified follows:
$G$ is finite iff $M(G)$ is finite, and in that case $M(G)$ is the same as defined in § 7.
$G$ is finite iff $M(G)$ is torsion-less (= free), and $height(G)=dim M(G)$.
For any perfect extension $K/k$, there is a functorial isomorphism $M(G\otimes_k K)\simeq W(K)\otimes_{W(k)} M(G)$.
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