Remarks on $K(1)$-local $K$-theory

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 Künneth formula in $K(1)$-local $K$-theory for adding $p$-power roots of unity to $R$.

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