Purity in chromatically localized algebraic $K$-theory
| Autor: | Land, Markus, Mathew, Akhil, Meier, Lennart, Tamme, Georg |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | J. Amer. Math. Soc. 37 (2024), 1011-1040 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1090/jams/1043 |
| Popis: | 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 $\mathrm{TC}(\tau_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$. Comment: v5: 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 |
| Databáze: | arXiv |
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