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

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By a similar discussion (replacing pp by FF) as in § 8, we have:

Remark

If GG is is a connected finite type formal group, define

M(G):=limM(kerF G n)M(G):= lim M(ker F^n_G)

This is a module over the FF-completion D^ k\hat D_k of D kD_k.

Theorem

The Dieudonné-functor is an equivalence

{FftcD^ kMod fin.len.quot GM(G)\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 D^ k\hat D_k-modules MM such that M/FMM/FM has finite length. Moreover we have:

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

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

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