On the Residue Sequence in Logarithmic Topological Cyclic Homology

Abstract

As a localizing invariant, THH participates in localization sequences of cyclotomic spectra. We resolve a conjecture of Rognes by relating these to residue sequences in logarithmic THH. Consequently, logarithmic THH, TR, and TC serve as strict generalizations of the constructions of Hesselholt--Madsen and Blumberg--Mandell, which moreover enjoy localization sequences without the regularity hypotheses usually required for dévissage. Combined with work of Ramzi--Sosnilo--Winges, our results imply that there exists a stable infinity-category C such that THH(C), TR(C), and TC(C) realize the relevant logarithmic term for specific log structures, such as the natural ones on discrete valuation rings, connective complex K-theory, and truncated Brown--Peterson spectra. Finally, we conjecture that the category C can be chosen to reflect the additional structure present on the logarithmic terms, and we give evidence for this in the case of discrete valuation rings.

arXiv comment29 pages, comments welcome!