Stable flatness of nonarchimedean hyperenveloping algebras

Let L be a completely valued nonarchimedean field and g a finite-dimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) agrees with its Arens–Michael envelope and, furthermore, is a stably flat completion of its universal enveloping algebra. As an application, we prove that the Taylor relative cohomology for the locally convex algebra F(g) is naturally isomorphic to the Lie algebra cohomology of g.