Computations of relative topological coHochschild homology

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