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Thomason filtration via $T(1)$-local TC

Hyungseop Kim
Vol. 10 (2025), No. 1, 1–54
DOI: 10.2140/akt.2025.10.1
Abstract

We construct a natural filtration on T(1)-local topological cyclic homology for any animated commutative rings using prismatic cohomology and descent theory. In the course of the construction, we also study some general properties of prismatic cohomology complexes over perfect prisms after inverting distinguished generators. The construction is intrinsic to topological cyclic homology and recovers Thomason’s spectral sequence for T(1)-local algebraic K-theory via the cyclotomic trace map; as a consequence, we also recover the étale comparison for prismatic cohomology.

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Keywords
algebraic \hboxK-theory, topological cyclic homology, motivic filtration, prismatic cohomology, étale comparison
Mathematical Subject Classification
Primary: 19D55
Secondary: 14F30, 19E20
Milestones
Received: 12 April 2024
Revised: 15 December 2024
Accepted: 6 January 2025
Published: 4 February 2025
Authors
Hyungseop Kim
CNRS, Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay
Orsay
France
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).