The étale symmetric Künneth theorem

Abstract

Let k be an algebraically closed field, lne {{,textrm{char},}}k a prime number, and X a quasi-projective scheme over k. We show that the étale homotopy type of the dth symmetric power of X is textbf{Z}/l-homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the textbf{Z}/l-local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.