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arXiv 2001.10425math.KTmath.AT
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色散局部化代数K-理论中的纯性

Purity in chromatically localized algebraic $K$-theory

Markus Land, Akhil Mathew, Lennart Meier, Georg Tamme

备注：
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
发表信息：
J. Amer. Math. Soc. 37 (2024), 1011-1040
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AI总结：

本文证明了环谱望远镜局部化代数K-理论中的纯性性质，指出K(R)的T(n)-局部化仅依赖于R的特定局部化，并推导出多个消没结果与等价关系。

摘要：

我们证明了环谱的望远镜局部化代数K-理论中的一个纯性性质：对于n≥1，K(R)的T(n)-局部化仅依赖于R的T(0)⊕...⊕T(n)-局部化。这补充了Waldhausen在有理K-理论中的经典结果。将我们的结果与Clausen--Mathew--Naumann--Noel的工作相结合，可以发现对于n≥1，L_{T(n)}K(R)事实上仅依赖于R的T(n-1)⊕T(n)-局部化。作为推论，我们推导出了望远镜局部化K-理论的几个消没结果，以及对于n≥2，在T(n)-局部化之后K(R)与TC(τ_{≥0} R)之间的等价关系。

英文摘要：

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 $n\geq 2$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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