This paper explores the duality relationship between the multiplicative $K$-theory of a Banach algebra $A$ and the indecomposables in the smooth cohomology of the associated general linear group. The main technical result is that a spectral sequence of coprimitive Hopf algebras induces a long exact sequence of indecomposables. This is applied to the Van Est spectral sequence yielding the above formal duality relationship. In the last section we also reinterpret the Van Est spectral sequence as a Serre spectral sequence in continuous cohomology.