Purity in Chromatically Localized Algebraic K-theory

Abstract

We prove a purity property in telescopically localized algebraic K-theory of ring spectra: For n ≥ 1, the T(n)-localization of K(R) only depends on the (Formula Presented)-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 (Formula Presented)-localization of R, again for n ≥ 1. As consequences, we deduce several vanishing results for telescopically localized K-theory, as well as an equivalence between K(R) and TC (T≥0)R after T(n)-localization for n≥2.