Topological Hochschild Homology of $K/p$ as a $K_p^\wedge$ module

Abstract

Let $R$ be an $E_\infty$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BGL_1R$, one can construct a Thom spectrum, $X^\zeta$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \Omega Y$) and $\zeta$ is homotopy equivalent to $\Omega f$ for a map $f$ from $Y$ to $B^2GL_1R$, then the Thom spectrum has an $A_\infty$-ring structure. The Topological Hochschild Homology of these $A_\infty$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$. This paper considers the case $X=S^1$, $R=K_p^\wedge$, the p-adic $K$-theory spectrum, and $\zeta = 1-p \in \pi_1BGL_1K_p^\wedge$. The associated Thom spectrum $(S^1)^\zeta$ is equivalent to the mod p $K$-theory spectrum $K/p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A_\infty$-ring structure. I will compute $\pi_*THH^{K_p^\wedge}(K/p)$ using its description as a Thom spectrum.