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

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.