\'Etale Covers and Local Algebraic Fundamental Groups

Abstract

Let $X$ be a normal noetherian scheme and $Z \subseteq X$ a closed subset of codimension $\geq 2$. We consider here the local obstructions to the map $\hat{\pi}_{1}(X\backslash Z) \to \hat{\pi}_{1}(X)$ being an isomorphism. Assuming $X$ has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-\'{e}tale cover of $X$, where the corresponding map on fundamental groups is an isomorphism.