论文标题
纯度在色度局部代数$ k $ - 理论中
Purity in chromatically localized algebraic $K$-theory
论文作者
论文摘要
我们证明，在望远镜本地化的代数$ k $的纯度属性中，环光谱理论：对于$ n \ geq 1 $，$ t（n）$ - $ k（r）$的本地化仅取决于$ t（0）\ oplus \ oplus \ dots \ oplus \ oplus \ oplus t（n）$ - $ r $的本地化。这补充了Waldhausen在Rational $ K $ - 理论中的经典结果。将我们的结果与Clausen的作品结合在一起 - 大师 - Naumann- noel，发现$ l_ {t（n）} k（r）$实际上仅取决于$ t（n-1）\ oplus t（n）$ - $ r $的本地化，又是$ n \ geq 1 $。结果，我们推断出望远镜本地化的$ k $ - 理论的几个消失结果，以及$ k（r）$和$ \ mathrm {tc}（τ_ {\ geq 0} r）$ t（n）$之后的等价性 - $ n \ geq 2 $。
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 $n\geq 2$.
