On topological coHochschild homology and cotensor products

TL;DR

This paper investigates the cotensor product of comodules in the context of $ ext{Mod}_R$ for an $ ext{E}_ ext{infty}$-ring spectrum, and applies these findings to analyze the structure and invariance properties of topological coHochschild homology.

Contribution

It introduces new results on cotensor products in $ ext{Mod}_R$ and demonstrates Morita-Takeuchi invariance of topological coHochschild homology, extending duality principles from topological Hochschild homology.

Findings

Cotensor product of comodules studied in $ ext{Mod}_R$ for $ ext{E}_ ext{infty}$-ring spectra.

Established Morita-Takeuchi invariance of topological coHochschild homology.

Duality between properties of coTHH and THH confirmed.

Abstract

In this work, we first study the cotensor product of comodules in the $\infty$-category $\mathrm{Mod}_R$ for a connected $\mathbb{E}_{\infty}$-ring spectrum $R$. We then apply these results to analyze higher coalgebra structures of topological coHochschild homology (coTHH) and establish its Morita-Takeuchi invariance, which are precisely dual to the corresponding properties of topological Hochschild homology.