# 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

$M(G):= lim M(ker F^n_G)$

This is a module over the $F$-completion $\hat D_k$ of $D_k$.

###### Theorem

The Dieudonné-functor is an equivalence

$\begin{cases} Fftc\to \hat D_k-Mod_{fin.len.quot} \\ G\mapsto M(G) \end{cases}$

between the category of connected formal groups of finite type and the category of $\hat D_k$-modules $M$ such that $M/FM$ has finite length. Moreover we have:

1. $G$ is finite iff $M(G)$ has finite length iff $F^n M(G)=0$ for large $n$.

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

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