The étale symmetric Künneth theorem

Abstract

Let $k$ be an algebraically closed field, $l\ne \mathrm{char}k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the étale 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 étale homotopy type of $X$. We deduce that the $\mathbb{Z}/l$-local étale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.

arXiv commentadded Prop. 3.7 about localizations of infinity-categories of pro-objects. Final version, to appear in Math. Z