Purity in chromatically localized algebraic $K$-theory
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866910544935518208 |
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| author | Land, Markus Mathew, Akhil Meier, Lennart Tamme, Georg |
| author_facet | Land, Markus Mathew, Akhil Meier, Lennart Tamme, Georg |
| contents | 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 $\mathrm{TC}(τ_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_10425 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Purity in chromatically localized algebraic $K$-theory Land, Markus Mathew, Akhil Meier, Lennart Tamme, Georg K-Theory and Homology Algebraic Topology 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 $\mathrm{TC}(τ_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$. |
| title | Purity in chromatically localized algebraic $K$-theory |
| topic | K-Theory and Homology Algebraic Topology |
| url | https://arxiv.org/abs/2001.10425 |