Localization sequences for logarithmic topological cyclic homology

TL;DR

This paper introduces logarithmic variants of topological Hochschild and cyclic homology for E_k-rings with prelogarithmic structures, establishing localization sequences and computing examples like non-negative even periodic sphere spectra.

Contribution

It extends the theory of topological cyclic homology by incorporating logarithmic structures and provides new computational tools and examples.

Findings

Established localization sequences for log THH and log TC.

Computed log THH and log TC for non-negative even periodic sphere spectra.

Identified the fraction field of topological K-theory within this framework.

Abstract

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.