On the Residue Sequence in Logarithmic Topological Cyclic Homology

TL;DR

This paper advances the understanding of logarithmic topological cyclic homology by establishing new residue sequences, generalizing previous constructions, and connecting these to specific log structures in stable infinity-categories.

Contribution

It resolves a conjecture relating residue sequences in logarithmic THH to cyclotomic spectra, generalizes existing constructions, and links these to stable infinity-categories for key examples.

Findings

Logarithmic THH, TR, and TC generalize previous constructions.

Localization sequences are established without regularity hypotheses.

Existence of a stable infinity-category realizing logarithmic terms for key log structures.

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 d\'evissage. 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.