Let $p\in \mathbb{Z}$ be an odd prime. We prove a spectral version of Tate-Poitou duality for the algebraic $K$-theory spectra of number rings with $p$ inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $K(\mathcal{O}_{F})^{\scriptscriptstyle\wedge}_{p} \to TC(\mathcal{O}_{F})^{\wedge}_{p}$ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic $K$-theory of $\mathbb{Z}$.

中文翻译：

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K 理论 Tate-Poitou 对偶性和分圆轨迹的纤维

让 $p\in \mathbb{Z}$ 是一个奇素数。我们证明了 Tate-Poitou 对偶性的谱版本，用于 $p$ 倒置的数环的代数 $K$ 理论谱。这确定了分圆轨迹的纤维的同伦类型 $K(\mathcal{O}_{F})^{\scriptscriptstyle\wedge}_{p} \to TC(\mathcal{O}_{F}) ^{\wedge}_{p}$ 在采取适当的连接覆盖后。作为一个应用，我们根据 $\mathbb{Z}$ 的代数 $K$ 理论确定球谱的分圆轨迹的同伦纤维奇素数处的同伦类型。