Remarks on $K(1)$-local $K$-theory
- Resource Type
- Working Paper
- Authors
- Bhatt, Bhargav; Clausen, Dustin; Mathew, Akhil
- Source
- Subject
- Mathematics - K-Theory and Homology
Mathematics - Algebraic Geometry
Mathematics - Algebraic Topology
- Language
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 Land--Meier--Tamme 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 $\mathrm{TC}$. Our second result yields a K\"unneth formula in $K(1)$-local $K$-theory for adding $p$-power roots of unity to $R$.
Comment: 13 pages, revised and final version