# Computational Tools for Topological CoHochschild Homology

## TL;DR

This paper develops computational tools for topological coHochschild homology, including a Hochschild-Kostant-Rosenberg type theorem and a spectral sequence, enabling new calculations for coalgebra spectra.

## Contribution

It introduces new computational methods, such as a spectral sequence and a theorem, to study topological coHochschild homology for coalgebras.

## Key findings

- Proved a Hochschild-Kostant-Rosenberg type theorem for differential graded coalgebras.
- Developed a coB"okstedt spectral sequence for computing coTHH homology.
- Performed several explicit computations using the spectral sequence.

## Abstract

In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coB\"okstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.