  Hess and Shipley defined an invariant of coalgebra spectra called topological
coHochschild homology, and Bohmann-Gerhardt-H{\o}genhaven-Shipley-Ziegenhagen
developed a coB\"okstedt spectral sequence to compute the homology of coTHH for
coalgebras over the sphere spectrum. We construct a relative coB\"okstedt
spectral sequence to study coTHH of coalgebra spectra over any commutative ring
spectrum $R$. Further, we use algebraic structures in this spectral sequence to
complete some calculations of the homotopy groups of relative topological
coHochschild homology.
