On -local

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 $\mathrm {TR}$ .

$K$-Theory of Cuspidal Curves Over a Perfectoid Base And Formal Analogues

“…Remark 1.3. The proof of this statement uses the Henselian ridgidity properties of nil invariant K-theory proven in [CMM21] to reduce to the same statement for topological cyclic homology, and then use the syntomic complexes Z p (i)(R⟨t⟩) from [BMS19] and the ku ∧ p -module structure from [Mat21,Example 5.5]. Elmanto has shown in [Elm21] that we should not expect affine invariance for topological cyclic homology.…”