Topological CoHochschild Homology and Thom Spectra

For an $\text{E}$-ring spectrum $R$ and a map $f:X\to Pic(R)$ of spaces, the Thom spectrum $\text{T}f$ is a comodule over $R\otimes \text{Si}X$. In this parper we study the topological coHochschild homology of $R\otimes \text{Si}X$ with coefficient $\text{T}f$. More concretely, for a simply connected space $X$, we will give a filtration on ${\mathrm{c}\mathrm{o}\mathrm{T}\mathrm{H}\mathrm{H}}^R(\text{T}f;R\otimes \text{Si}X)$ via the cellular structure of $X$. Furthermore, we will reduce the computation of ${\mathrm{c}\mathrm{o}\mathrm{T}\mathrm{H}\mathrm{H}}^R(\text{T}f;R\otimes \text{Si}X)$ to that of ${\mathrm{c}\mathrm{o}\mathrm{T}\mathrm{H}\mathrm{H}}^R(R\otimes \text{Si}G)$ for some group-like ${\mathbb{E}}_1$-spaces. Finally, we will use these results to study properties of $\mathrm{c}\mathrm{o}\mathrm{T}\mathrm{H}\mathrm{H}(\text{T}f;R\otimes \text{Si}X)$.