Localization sequences for logarithmic topological cyclic homology

We introduce the notion of an E_k-ring with prelogarithmic structure, define logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this context, and establish localization sequences for these theories.

Our approach is based on Thom R-algebras.

It extends and strengthens our earlier work on the subject in several regards.

Our examples include the fraction field of topological K-theory, the existence of which was suggested by calculations by Ausoni and the first author.

To illustrate the computational accessibility of log THH and log TC, we determine these for non-negative even periodic sphere spectra, with their canonical prelogarithmic structures.