乘积的完备化
Profinite completions of products
AI总结：
本文提出一个准则，证明完备化保持qcqs概形的étale同调类型的积，填补了Künneth公式的证明空白。
摘要：
范畴论和代数拓扑中的一大难点在于，完备化函子不保持有限积。在本文中，我们提供了一个新的、可检验的准则，用于保证pro空间X和Y的完备化满足X×Y的完备化等于X和Y的完备化积。利用这一准则，我们证明了完备化保持qcqs概形的étale同调类型的积。这填补了Chough证明分离闭字段上proper概形积的étale同调类型的Künneth公式中的空白。
英文摘要：
A source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. In this note, we provide a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the profinite completion of $X\times Y$ agrees with the product of the profinite completions of $X$ and $Y$. Using this criterion, we show that profinite completion preserves products of étale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the Künneth formula for the étale homotopy type of a product of proper schemes over a separably closed field.
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。
