The symmetric power and étale realisation functors commute

Abstract

We show that under mild hypotheses on a proper algebraic space $X$, the functors of taking its symmetric powers and its étale realisation commute up to weak equivalence. We conclude an effective version of the Dold-Thom theorem for the étale site and discuss the stabilisation results for the natural morphisms of étale homotopy groups ${\pi }_k{\mathrm{Sym}}^{n}X\to {\pi }_k{\mathrm{Sym}}^{n+1}X$ in the context of the Weil conjectures.