A comparison of locally analytic group cohomology and Lie algebra cohomology for p-adic Lie groups

TL;DR

This paper provides a new proof and generalization of Lazard's comparison theorem, establishing an isomorphism between locally analytic group cohomology and Lie algebra cohomology for p-adic Lie groups using formal group laws.

Contribution

It introduces a novel proof independent of Lazard's original approach and extends the comparison to broader classes of p-adic Lie groups.

Findings

The Lazard morphism is an isomorphism for an open subgroup of G.

The proof employs formal group law theory.

The result applies to K-Lie groups over finite extensions of p-adic numbers.

Abstract

The main result of this work is a new proof and generalization of Lazard's comparison theorem of locally analytic group cohomology with Lie algebra cohomology for K-Lie groups, where K is a finite extension of the p-adic numbers. We show the following theorem: Let K be a finite extension of the p-adic numbers and let G be a K-Lie group. Then there exists an open subgroup U of G such that the Lazard morphism, which is induced by differentiating cochains, is an isomorphism. The proof of this theorem is independent of the proof of Lazard's comparison result. Our strategy to prove the comparison isomorphism between locally analytic group cohomology and Lie algebra cohomology uses the theory of formal group laws. And in a second step we consider standard groups.