Lazard isomorphism ($H_c^{\ast }(G,M)\cong H^{\ast }(\mathfrak{g},M)$)

The Lazard isomorphism compares continuous group cohomology of suitable $p$-adic analytic groups with Lie algebra cohomology of their associated Lie algebras. It converts a cohomological question about a nonlinear multiplication law into a calculation with alternating multilinear maps. The hypotheses matter: one cannot assert this comparison for every topological group.

One precise integral version takes a uniform pro-p group $G$ and a finite free ${\mathbb{Z}}_p$-module $M$. For odd $p$, require the continuous action to have image in $1+p\mathrm{End}(M)$; for $p=2$, use $1+4\mathrm{End}(M)$. With $\mathfrak{g}$ the associated integral Lazard Lie algebra,This is an integral refinement of the rational comparison.^[ref]

For the additive group $G={\mathbb{Z}}_p$ with trivial coefficients $M={\mathbb{Z}}_p$, continuous one-cocycles are $x\mapsto ax$. The associated rank-one abelian Lie algebra has $H^1\cong {\mathrm{Hom}}_{{\mathbb{Z}}_p}({\mathbb{Z}}_p,{\mathbb{Z}}_p)\cong {\mathbb{Z}}_p$, giving the same answer. Rationalizing coefficients alone would not establish an integral comparison in general.