Purity in chromatically localized algebraic $K$-theory

We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\ge 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \cdots \oplus T(n)$-localization of $R$. This complements a classical result of Waldhausen in rational $K$-theory. Combining our result with work of Clausen–Mathew–Naumann–Noel, one finds that $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $\mathrm{T}\mathrm{C}({\tau }_{\ge 0}R)$ after $T(n)$-localization for $n\ge 2$.