Purity in chromatically localized algebraic 𝐾-theory

Land, Markus; Mathew, Akhil; Meier, Lennart; Tamme, Georg

doi:10.1090/jams/1043

Purity in chromatically localized algebraic $K$-theory
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by Markus Land, Akhil Mathew, Lennart Meier and Georg Tamme;

J. Amer. Math. Soc. 37 (2024), 1011-1040

DOI: https://doi.org/10.1090/jams/1043

Published electronically: February 1, 2024

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Abstract:

We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\geq 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \dots \oplus T(n)$-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 $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $TC(\tau _{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$.

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