  In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for
an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra
$\mathcal{H}$ is bijective. We show that the cyclic module associated to the
diagonal of $A \natural \mathcal{H}$ is isomorphic with the cyclic module of
the crossed product algebra $A \rtimes \mathcal{H}$. This enables us to derive
a spectral sequence for the cyclic homology of the crossed product algebra. We
also construct a cocylindrical module for Hopf module coalgebras and establish
a similar spectral sequence to compute the cyclic cohomology of crossed product
coalgebras.
