A Note on Topological Hochschild Homology Relative to ${\text{Sphere}}_{W(k)}[x_0,x_1,\dots ,x_n]$

We explain the relation between the relative topological Hochschild homology $\text{THH}(R\mathrm{/}{\text{Sphere}}_{W(k)}[x_0,\dots ,x_n])$ and the Nygaard completed Frobenius twisted relative prismatic cohomology ${\widehat{\text{Prism}}}_{R\mathrm{/}W(k)[x_0,\dots ,x_n{]}^{\wedge }}^{(1)}$, where $W(k)[x_0,x_1,\dots ,x_n]\to R$ is relatively quasiregular semiperfectoid. As an application, for $R={\mathbb{Z}}_p[x]\mathrm{/}(px)$, we compute ${\pi }_{\ast }\text{THH}(R{)}_p^{\wedge }$ by descent along $\text{THH}(R{)}_p^{\wedge }\to \text{THH}(R\mathrm{/}{\text{Sphere}}_p[z,x])$, where $R={\mathbb{Z}}_p[x]\mathrm{/}(px)$ is regarded as an $\text{Einfty}$-${\text{Sphere}}_p[z,x]$-algebra through ${\text{Sphere}}_p[z,x]\stackrel{z\mapsto p,x\mapsto x}{\to }{\mathbb{Z}}_p[x]\mathrm{/}(px)$.