# nLab Demazure, lectures on p-divisible groups, III.9, Dieudonné modules (connected formal groups of finite type)

This entry is about a section of the text

By a similar discussion (replacing $p$ by $F$) as in § 8, we have:

###### Remark

If $G$ is is a connected finite type formal group, define

$M\left(G\right):=\mathrm{lim}M\left(\mathrm{ker}{F}_{G}^{n}\right)$M(G):= lim M(ker F^n_G)

This is a module over the $F$-completion ${\stackrel{^}{D}}_{k}$ of ${D}_{k}$.

###### Theorem

The Dieudonné-functor is an equivalence

$\left\{\begin{array}{l}\mathrm{Fftc}\to {\stackrel{^}{D}}_{k}-{\mathrm{Mod}}_{\mathrm{fin}.\mathrm{len}.\mathrm{quot}}\\ G↦M\left(G\right)\end{array}$\begin{cases} Fftc\to \hat D_k-Mod_{fin.len.quot} \\ G\mapsto M(G) \end{cases}

between the category of connected formal groups of finite type and the category of ${\stackrel{^}{D}}_{k}$-modules $M$ such that $M/\mathrm{FM}$ has finite length. Moreover we have:

1. $G$ is finite iff $M\left(G\right)$ has finite length iff ${F}^{n}M\left(G\right)=0$ for large $n$.

2. $G$ is smooth iff $F:M\left(G\right)\to M\left(G\right)$ is injective. In that case $\mathrm{dim}\left(G\right)=\mathrm{length}\left(M\left(G\right)/\mathrm{FM}\left(G\right)\right)$.

Revised on May 27, 2012 13:49:18 by Stephan Alexander Spahn (79.227.168.80)