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My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$).

Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action.

Lazard showed that, under certain hypothesis, the continuous cohomology of $G$ in $M$ is isomorphic to (a sub-module of) the cohomology of its Lie algebra in $M$.

My question is now : what if $M$ has coefficients in $\mathbb{F}_p$ ? Is there a way to compute the continuous cohomology of $G$ in $M$ using its Lie algebra ?

Thanks in advance.

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Quite late, but the following paper may contain the answer to your question:

A. Huber, G. Kings and N. Naumann. Some complements to the Lazard isomorphism. Compositio Mathematica 147, 2011, 235-262.

See here for the arXiv version.

One of the results of the paper is an integral version of Lazard's isomorphism under some technical assumptions that you will have to look up in the paper.

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