1 results
On
$K(1)$-local 
$\mathrm {TR}$
- Part of
-
- Journal:
- Compositio Mathematica / Volume 157 / Issue 5 / May 2021
- Published online by Cambridge University Press:
- 04 May 2021, pp. 1079-1119
- Print publication:
- May 2021
-
- Article
- Export citation
-
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 
$\mathrm {TR}$.
