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 $\mathrm {TR}$.

我们讨论$ \ mathrm {TR} $及其$ K（1）$-本地化的一些常规属性。我们证明后$ K（1）$ -localization，$ \ mathrm {TR} $的$ H \ mathbb {Z} $ -代数是在土地Tamme意义上的截断不变，并推断$ H $ -descent结果。我们表明，对于具有混合特征的正则环，$ \ mathrm {TR} $渐近为$ K（1）$ -局部，扩展了Hesselholt和Madsen的结果。作为这些方法的应用以及环原子光谱理论的最新进展，我们构建了与$ K（1）$-局部$ K $相关的Thomason光谱序列的类似物。- $ K（1）$-局部$ \ mathrm {TR} $的理论和étale同调。