  Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$.
The group of invertible relative Hopf modules maps to the Picard group of $A$,
and the kernel is described as a quotient group of the group of invertible
grouplike elements of the coring $A\ot H$, or as a Harrison cohomology group.
Our methods are based on elementary $K$-theory. The Hilbert 90 Theorem follows
as a corollary. The part of the Picard group of the coinvariants that becomes
trivial after base extension embeds in the Harrison cohomology group, and the
image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a
cosemisimple Hopf algebra over a field.
