A Note on Topological Hochschild Homology Relative to $\Sphere_{W(k)}[x_0,x_1,\ldots,x_n]$

Abstract

We explain the relation between the relative topological Hochschild homology $\THH(R/\Sphere_{W(k)}[x_0,\ldots,x_n])$ and the Nygaard completed Frobenius twisted relative prismatic cohomology $\widehat{\Prism}^{(1)}_{R/W(k)[x_0,\ldots,x_n]^\wedge}$, where $W(k)[x_0,x_1,\ldots,x_n]\rightarrow R$ is relatively quasiregular semiperfectoid. As an application, for $R=\Z_p[x]/(px)$, we compute $π_*\THH(R)^\wedge_p$ by descent along $\THH(R)^\wedge_p\rightarrow \THH(R/\Sphere_p[z,x])$, where $R=\Z_p[x]/(px)$ is regarded as an $\Einfty$-$\Sphere_p[z,x]$-algebra through $\Sphere_p[z,x]\xrightarrow{z\mapsto p,x\mapsto x}\Z_p[x]/(px)$.