Some complements to the Lazard isomorphism

TL;DR

This paper refines Lazard's isomorphism by extending it to integral coefficients under certain conditions and demonstrating its realization through differentiation of locally analytic cochains for algebraic groups.

Contribution

It extends Lazard's isomorphism to integral coefficients and shows its realization via differentiation for algebraic groups.

Findings

Isomorphism extended to integral coefficients under specific conditions

Realization of the isomorphism through differentiation of locally analytic cochains

Enhanced understanding of cohomological relationships in p-adic Lie groups

Abstract

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.