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

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

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 $dim(G)= length(M(G)/ FM(G))$.