Cyclic Cohomology of Crossed Coproduct Coalgebras

We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We introduce the cocylindrical module $C \natural^{} \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra with bijective antipode and $C$ is a Hopf comodule coalgebra over $\mathcal{H}$. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra $C > \blacktriangleleft \mathcal{H}$ and $\Delta(C \natural^{}\mathcal{H})$, the cocyclic module related to the diagonal of $C \natural^{} \mathcal{H}$. We approximate $HC^{\bullet}(C > \blacktriangleleft \mathcal{H})$ by a spectral sequence and we give an interpretation for $\mathsf{E}^0, \mathsf{E}^1$ and $\mathsf{E}^2$ terms of this spectral sequence.