The symmetric power and \'{e}tale 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 \'{e}tale realisation commute up to weak equivalence. We conclude an effective version of the Dold-Thom theorem for the \'{e}tale site and discuss the stabilisation results for the natural morphisms of \'{e}tale homotopy groups $\pi_k \mathrm{Sym}^n X \to \pi_k \mathrm{Sym}^{n+1} X$ in the context of the Weil conjectures.