The pro-étale fundamental group of singular schemes

Abstract

We compute the pro-étale fundamental group of a connected Nagata J-2 scheme in terms of the étale fundamental groups of the normalizations of its irreducible components and a discrete free group. The result generalizes a formula of E. Lavanda for semi-stable curves and relies on a combination of proper descent techniques for étale morphisms and a combinatorial van Kampen construction for Noohi groups. As a by-product we characterize when a continuous representation of the pro-étale fundamental group factors through a discrete quotient.