Universal covering spaces and fundamental groups in algebraic geometry as schemes

TL;DR

This paper develops a scheme-theoretic framework for universal covers and fundamental groups in algebraic geometry, extending topological concepts to schemes and connecting to étale fundamental groups.

Contribution

It introduces a universal cover and fundamental group family as schemes, generalizing classical topological notions to algebraic geometry.

Findings

Constructed a fundamental group family as schemes for all connected quasicompact quasiseparated schemes.

Showed a geometric fiber of this family is the étale fundamental group.

Connected the new construction to Deligne's prior work using different methods.

Abstract

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the 'etale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.