Computations of relative topological coHochschild homology

TL;DR

This paper develops a spectral sequence framework for computing the relative topological coHochschild homology of coalgebra spectra over any commutative ring spectrum, extending previous methods and enabling new calculations.

Contribution

It introduces a relative coB"okstedt spectral sequence for coalgebra spectra over arbitrary commutative ring spectra, generalizing earlier approaches.

Findings

Constructed a new spectral sequence for relative coTHH.

Applied algebraic structures to compute homotopy groups.

Extended computations to broader classes of coalgebra spectra.

Abstract

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.