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Remarks on \(K(1)\)-local \(K\)-theory. (English) Zbl 1454.19004

Summary: We prove two basic structural properties of the algebraic \(K\)-theory of rings after \(K(1)\)-localization at an implicit prime \(p\). Our first result (also recently obtained by M. Land et al. [“Vanishing results for chromatic localizations of algebraic \(K\)-theory”, Preprint. arXiv:2001.10425] by different methods) states that \(L_{K(1)} K(R)\) is insensitive to inverting \(p\) on \(R\); we deduce this from recent advances in prismatic cohomology and TC. Our second result yields a Künneth formula in \(K(1)\)-local \(K\)-theory for adding \(p\)-power roots of unity to \(R\).

MSC:

19D50 Computations of higher \(K\)-theory of rings
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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