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On $K(1)$-local $\mathrm {TR}$

Part of: Higher algebraic $K$-theory Homotopy theory

Published online by Cambridge University Press:  04 May 2021

Akhil Mathew
Akhil Mathew*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL60605, USAamathew@math.uchicago.edu
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Abstract

We discuss some general properties of $\mathrm {TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm {TR}$ of $H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm {TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and étale cohomology for $K(1)$-local $\mathrm {TR}$.

Keywords

Algebraic K-theorytopological cyclic homologyLichtenbaum–Quillen conjecture

MSC classification

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology
Secondary: 55P42: Stable homotopy theory, spectra
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 5 , May 2021 , pp. 1079 - 1119
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Copyright
© The Author(s) 2021

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Footnotes

This work was done while the author was a Clay Research Fellow.

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