On the Residue Sequence in Logarithmic Topological Cyclic Homology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913898784882688 |
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| author | Lundemo, Tommy |
| author_facet | Lundemo, Tommy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14545 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Residue Sequence in Logarithmic Topological Cyclic Homology Lundemo, Tommy Algebraic Topology Algebraic Geometry K-Theory and Homology 55P43, 14F10, 19D55 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. |
| title | On the Residue Sequence in Logarithmic Topological Cyclic Homology |
| topic | Algebraic Topology Algebraic Geometry K-Theory and Homology 55P43, 14F10, 19D55 |
| url | https://arxiv.org/abs/2506.14545 |