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 $C{\mathrm{♮}}^{}\mathcal{H}$, 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 $C{\mathrm{♮}}^{}\mathcal{H}$. We approximate $H{C}^{\bullet }(C>◀\mathcal{H})$ 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.