  Lazard showed in his seminal work "Groupes analytiques $p$-adiques" that for
rational coefficients continuous group cohomology of $p$-adic Lie-groups is
isomorphic to Lie-algebra cohomology. We refine this result in two directions:
firstly we extend his isomorphism under certain conditions to integral
coefficients and secondly, we show that for algebraic groups, his isomorphism
can be realized by differentiating locally analytic cochains.
