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 T(0)⊕⋯⊕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 LT(n)K(R) in fact only depends on the T(n−1)⊕T(n)-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(τ≥0R) after T(n)-localization for n≥2