# p-adic Lie group vs Lie algebra cohomology with mod p coefficients

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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## 1 Answer

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.

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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