On schemes with trivial higher étale homotopy groups

Abstract

Let $(X,\bar{x})$ be a pointed connected noetherian scheme. In this note, we give characterizations for the vanishing of the second étale homotopy group ${\pi }_2^{\acute{\mathrm{e}}\mathrm{t}}(X,\bar{x})$ in terms of splitting profinite-étale covers of $X$, and by means of universal covering spaces of the Artin-Mazur-Friedlander étale homotopy type $Et(X)$. In particular, this provides certain classes of schemes for which the Brauer map is surjective.