We prove a purity property in telescopically localized algebraic K K -theory of ring spectra: For n ≥ 1 n\geq 1 , the T ( n ) T(n) -localization of K ( R ) K(R) only depends on the T ( 0 ) ⊕ ⋯ ⊕ T ( n ) T(0)\oplus \dots \oplus T(n) -localization of R R . This complements a classical result of Waldhausen in rational K K -theory. Combining our result with work of Clausen–Mathew–Naumann–Noel, one finds that L T ( n ) K ( R ) L_{T(n)}K(R) in fact only depends on the T ( n − 1 ) ⊕ T ( n ) T(n-1)\oplus T(n) -localization of R R , again for n ≥ 1 n \geq 1 . As consequences, we deduce several vanishing results for telescopically localized K K -theory, as well as an equivalence between K ( R ) K(R) and T C ( τ ≥ 0 R ) TC(\tau _{\geq 0} R) after T ( n ) T(n) -localization for n ≥ 2 n\geq 2 .

我们证明了望远镜局域代数 K K -环谱理论中的纯度性质：对于 n ≥ 1 n\geq 1 ， K ( R ) K(R) 的 T ( n ) T(n) -局域化仅取决于 T ( 0 ) ⊕ ⋯ ⊕ T ( n ) T(0)\oplus \dots \oplus T(n) - R R 的定位。这补充了 Waldhausen 在理性 KK 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，我们发现 L T ( n ) K ( R ) L_{T(n)}K(R) 实际上仅取决于 T ( n − 1 ) ⊕ T ( n ) T(n-1)\oplus T(n) - R R 的定位，同样对于 n ≥ 1 n \geq 1 。结果，我们推导出伸缩局域 K K 理论的几个消失结果，以及 K ( R ) K(R) 和 T C ( τ ≥ 0 R ) TC(\tau _{\geq 0} R) 之间的等价关系T ( n ) T(n) - n ≥ 2 n\geq 2 的定位。