Purity in chromatically localized algebraic $K$-theory

Abstract

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

arXiv commentv5: accepted version; v4:new introduction, updated references, 26 pages; v3: New author, new title; this is an almost completely rewritten version of the paper that was previously entitled `Vanishing results for chromatic localizations of algebraic K-theory'. In particular, we affirmatively answer a question about purity for telescopically localized algebraic K-theory from the previous version

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• Descent and cyclotomic redshift for chromatically localized algebraic K-theory
• Algebraic K-theory of elliptic cohomology
• Segal K-theory of vector spaces with an automorphism
• On the Non-Existence of J-Homomorphisms of Higher Height
• Thomason filtration via $T(1)$-local $\mathrm{TC}$