\'Etale Covers and Fundamental Groups of Schematic Finite Spaces

TL;DR

This paper develops a Galois theory for finite étale covers of schematic finite spaces, defining an étale fundamental group that aligns with classical theories for schemes, using cohomological properties of these spaces.

Contribution

It introduces a Galois category framework for finite étale covers of schematic finite spaces and connects it with classical étale fundamental groups of schemes.

Findings

Defined the category of finite étale covers as a Galois category.

Established the étale fundamental group for schematic finite spaces.

Showed equivalence with classical étale fundamental groups for schemes.

Abstract

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