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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

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. $G=\cup \ker p^n{\mathrm{id}}_G$

2. $\ker p{\mathrm{id}}_G$ 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+1}\stackrel{p^n}{\to}ker p^{n+1}

$$0\to \ker p^n\to \ker p^{n+m}\stackrel{p^n}{\to }\ker p^m$$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 $M(G)=\mathrm{colim}M(\ker p^n)$

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

$$F_M(\mathrm{wm})=w^{(p)}{F}_M(m)$$F_M(wm)=w^{(p)}F_M (m)

$$V_M(w_m^{(p)})=w{V}_n(m)$$V_M(w^{(p)}_m)=w V_n(m)

$$F_M{V}_M=V_M{F}_M=p{\mathrm{id}}_M$$F_M V_M=V_M F_M=p id_M

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

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

Conversely if $M$ is as before we define $G:=\mathrm{colim}{G}_n$ 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 $\mathrm{height}(G)=\dim M(G)$.

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 $M(G^{\prime })={\mathrm{Mod}}_{W(k)}(M(G),W(k)$ with

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

and

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

(Demazure Theorem p.71-72)

Theorem

a) The Dieudonné functor

$$\{\begin{array}{l}{\mathrm{Torf}}_p\to (\mathrm{fin}W(k)-\mathrm{Mod},F,V)\\ G\to M(G)\end{array}$$\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

$$F_M(\lambda m)={\lambda }^p{F}_M(m)$$F_M(\lambda m)=\lambda^{p} F_M(m)

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

$$F_M{V}_M=V_M{F}_M=p\cdot {\mathrm{id}}_M$$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 $\mathrm{colim}{G}_n$ 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 $\mathrm{height}(G)=\dim M(G)$.

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 $M(G^{\prime })={\mathrm{Mod}}_{W(k)}M(G)$, with $F_{M(G^{\prime })}f)(m)=f(V_{M}m{)}^{(p)}$ and $(V_{M(G^{\prime })}f)(m)=f(F_{M}m{)}^{(p^{(-1)}}$.

References

Michel Demazure, lectures on p-divisible groups web