Purity in chromatically localized algebraic 𝐾-theory

Abstract

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