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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

• 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

$$M(G):=\lim M(\ker F_G^n)$$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

$$\{\begin{array}{l}\mathrm{Fftc}\to {\hat{D}}_k-{\mathrm{Mod}}_{\mathrm{fin}.\mathrm{len}.\mathrm{quot}}\\ G\mapsto M(G)\end{array}$$\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 ${\hat{D}}_k$-modules $M$ such that $M/\mathrm{FM}$ 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 $\dim (G)=\mathrm{length}(M(G)/\mathrm{FM}(G))$.