\'Etale Covers and Local Algebraic Fundamental Groups

TL;DR

This paper investigates the local obstructions to the isomorphism of algebraic fundamental groups of a scheme minus a closed subset and the scheme itself, establishing conditions under which these obstructions are finite and relate to Galois quasi-étale covers.

Contribution

It proves the equivalence between finiteness of local obstructions and the existence of Galois quasi-étale covers under the assumption of a regular alteration.

Findings

Finiteness of local obstructions is equivalent to the existence of a Galois quasi-étale cover.

Establishes conditions for the isomorphism of fundamental groups after removing a codimension ≥ 2 subset.

Provides criteria for when the local algebraic fundamental group map is an isomorphism.

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.