标题： 有限群完成的乘积 显示英文标题

标题： Profinite completions of products

摘要： 在有限上同伦理论中，一个困难的来源是有限完成函子不保持有限乘积。 在本说明中，我们提供了对pro空间$X$和$Y$的新、可检查的条件，该条件保证$X\times Y$的有限完成与$X$和$Y$的有限完成的乘积一致。 利用这个条件，我们证明了有限完成保持qcqs概形的埃雷特同伦类型的乘积。 这填补了Chough在其关于分离闭域上有限概形乘积的埃雷特同伦类型的Künneth公式的证明中的一个漏洞。 显示英文摘要

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 \'{e}tale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the K\"{u}nneth formula for the \'{e}tale homotopy type of a product of proper schemes over a separably closed field.