K-理论 Tate-Poitou 对偶在素数 2 处
K-theoretic Tate-Poitou duality at prime 2
AI总结：
本文推广了 K-理论 Tate-Poitou 对偶至素数 2，处理实嵌入带来的复杂性，并用整数环的代数 K-理论确定了球谱 cyclotomic trace 同伦纤维在素数 2 处的同伦类型。
摘要：
我们将 Blumberg 和 Mandell 关于奇数素数处 K-理论 Tate-Poitou 对偶的结果进行了推广，该对偶作为经典算术 Tate-Poitou 对偶的谱精化。该对偶针对数域中 p-整数环的 K(1)-局部化代数 K-理论及其通过 Z_p-Anderson 对偶的完备化而表述。本文通过处理素数 2 的情况完善了这一图景，此时数域中的实嵌入引入了额外的复杂性。作为应用，我们确定了球谱的 cyclotomic trace 的同伦纤维在素数 2 处的同伦类型，用整数环的代数 K-理论加以刻画。
英文摘要：
We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized algebraic K-theory of the ring of $p$-integers in a number field and its completion using the $\mathbb{Z}_p$-Anderson duality. This paper completes the picture by addressing the prime 2, where the real embeddings of number fields introduce extra complexities. As an application, we identify the homotopy type at prime 2 of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of the integers.
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