The étale symmetric Künneth theorem

Abstract

AbstractLet k be an algebraically closed field, $$l\ne {{\,\textrm{char}\,}}k$$ l ≠ 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 $$\textbf{Z}/l$$ Z / l -homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the $$\textbf{Z}/l$$ Z / l -local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.