Euclidean-valued group cohomology is always reduced

Abstract

Let [Formula: see text] be a topological field and [Formula: see text] a countable discrete group. Then for any linear [Formula: see text]-action on a finite-dimensional vector space over [Formula: see text], the groups of coboundaries in the inhomogeneous bar resolution are closed in all degrees, and hence the cohomology is reduced in all degrees. This can be deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.