TY - UNPB
T1 - Purity in chromatically localized algebraic K-theory
AU - Land, Markus
AU - Mathew, Akhil
AU - Meier, F.L.M.
AU - Tamme, Georg
PY - 2020/1/28
Y1 - 2020/1/28
N2 - 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
AB - 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
U2 - 10.48550/arXiv.2001.10425
DO - 10.48550/arXiv.2001.10425
M3 - Preprint
SP - 1
EP - 29
BT - Purity in chromatically localized algebraic K-theory
PB - arXiv
ER -