Étale Covers and Fundamental Groups of Schematic Finite Spaces

Abstract

We introduce the category of finite étale covers of an arbitrary schematic space X and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If X is a finite model of a scheme S, we show that the resulting Galois theory on X coincides with the classical theory of finite étale covers on S, and therefore, we recover the classical étale fundamental group introduced by Grothendieck. To prove these results, it is crucial to find a suitable geometric notion of connectedness for schematic spaces and also to study their geometric points. We achieve these goals by means of the strong cohomological constraints enjoyed by schematic spaces.