The étale symmetric Künneth theorem

Abstract

Let k be an algebraically closed field, l≠chark 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 Z/l-homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the Z/l-local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.