Topological CoHochschild Homology and Thom Spectra

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