The pro-\'etale fundamental group of singular schemes

TL;DR

This paper computes the pro-étale fundamental group of certain singular schemes, generalizing previous formulas and characterizing representations, using descent techniques and a combinatorial van Kampen approach.

Contribution

It provides a new formula for the pro-étale fundamental group of Nagata J-2 schemes, extending Lavanda's work to more general singular schemes.

Findings

Derived a formula relating the pro-étale fundamental group to normalizations and free groups.

Characterized when continuous representations factor through discrete quotients.

Utilized descent techniques and van Kampen constructions for Noohi groups.

Abstract

We compute the pro-\'etale fundamental group of a connected Nagata J-2 scheme in terms of the \'etale 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 \'etale morphisms and a combinatorial van Kampen construction for Noohi groups. As a by-product we characterize when a continuous representation of the pro-\'etale fundamental group factors through a discrete quotient.