The étale symmetric Künneth theorem

Zusammenfassung

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