The \'etale symmetric K\"unneth theorem

Marc Hoyois

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arXiv · 1810.00351

The \'etale symmetric K\"unneth theorem

Marc Hoyois

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Abstract

Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the \'etale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the \'etale homotopy type of $X$. We deduce that the $\mathbb Z/l$-local \'etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.

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Marc Hoyois. 2018-09-30. The \'etale symmetric K\"unneth theorem. https://arxiv.org/abs/1810.00351

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