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

摘要

For commutative ring spectra $R$, one can construct a Thom spectrum for spaces over $BGL_1R$. This specialises to the classical Thom spectra for spherical fibrations in the case of the sphere spectrum. The construction is useful in detecting $A_infty$-structures: a loop space (up to homotopy) over $BGL_1R$ yields an $A_infty$-ring structure on the Thom spectrum. The topological Hochschild homology of these $A_infty$-ring spectra may be expressed as Thom spectra. This paper uses the identification of topological Hochschild homology of Thom spectra to make computations. Specifically, we take $R$ to be the $p$-adic $K$-theory spectrum and consider a certain map from $S^1$ to $BGL_1R$, so that the Thom spectrum is equivalent to the $extrm{mod}, p$ $K$-theory spectrum. We make computations at odd primes.