$\mathrm{THH}$ and base-change for Galois extensions of ring spectra

Abstract

We treat the question of base-change in $THH$ for faithful Galois extensions of ring spectra in the sense of Rognes. Given a faithful Galois extension $A\to B$ of ring spectra, we consider whether the map $THH\left(A\right){\otimes }_{A}B\to THH\left(B\right)$ is an equivalence. We reprove and extend positive results of Weibel and Geller, and McCarthy and Minasian, and offer new examples of Galois extensions for which base-change holds. We also provide a counterexample where base-change fails.