TY - JOUR

T1 - Purity in Chromatically Localized Algebraic K-theory

AU - Land, M

AU - Mathew, A

AU - Meier, L

AU - Tamme, G

N1 - Publisher Copyright:
© 2024, Centre d'Etudes Mongoles et Siberiennes. All rights reserved.

PY - 2024

Y1 - 2024

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 (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.

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 (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.

UR - https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=d7dz6a2i7wiom976oc9ff2iqvdhv8k5x&SrcAuth=WosAPI&KeyUT=WOS:001157099900001&DestLinkType=FullRecord&DestApp=WOS_CPL

UR - http://www.scopus.com/inward/record.url?scp=85205089936&partnerID=8YFLogxK

U2 - 10.1090/jams/1043

DO - 10.1090/jams/1043

M3 - Article

SN - 0894-0347

VL - 37

SP - 1011

EP - 1040

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

IS - 4

ER -