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Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be the unique power series such that $dF_w=w$ and $F_w(0)=0$. Let $F_w^2(X,Y) = F_w(X \oplus Y) - F_w(X) - F_w(Y)$. Say that $w$ is second kind if $F_w^2$ has bounded coefficients and that $F_w$ is exact if $F_w$ has bounded coefficients. The 1st de Rham cohomology group of $G$ is defined by $$H^1_{dR}(G)= \text{\{second kind forms\}} / \text{\{exact forms\}}.$$

Theorem: the group $H^1_{dR}(G)$ has dimension $h$, the height of $G$.

Question: where can I find a proof of this?

The above definitions and theorem are in pages 633-634 of Colmez' "Periodes $p$-adiques des varietes abeliennes" for example, and he refers to Fontaine's book "Groupes $p$-divisibles sur les corps locaux", but without giving a precise reference. Iovita also uses these definitions in "Formal sections and de Rham cohomology of semistable abelian varieties" and refers to chapter V of Katz' "Crystalline cohomology, Dieudonne modules and Jacobi sums". In either case, I can't say that the references have been very helpful.

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If you like, I could send you a manuscript that is under construction that includes a proof of this. – Neil Strickland Jan 17 '11 at 18:13
Is $K$ a local (hence $p$-adic) field? – fherzig Jan 17 '11 at 18:46
@Neil: that would be great, thank you! – Laurent Berger Jan 17 '11 at 18:48
@Florian: yes absolutely, I forgot to mention that. $K$ should be a finite extension of $Q_p$ – Laurent Berger Jan 17 '11 at 18:49
Neil, I am interested in a proof too, is that too much work to post the proof here or secret or is it possible for you to share it ? Thanks by advance. – A M Jan 17 '11 at 19:48
up vote 17 down vote accepted

I have put an updated copy of my formal groups notes here:

They are not really finished, but the relevant material is discussed in Section 18.

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