Étale Covers and Fundamental Groups of Schematic Finite Spaces

Abstract

[EN] We introduce the category of finite étale covers of an arbitraryschematic space X and show that, equipped with an appropriate naturalfiber 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 schemeS, we show that the resulting Galois theory on X coincides with theclassical theory of finite étale covers on S, and therefore, we recover the classical étale fundamental group introduced by Grothendieck. Toprove these results, it is crucial to find a suitable geometric notion ofconnectedness for schematic spaces and also to study their geometric points. We achieve these goals by means of the strong cohomologicalconstraints enjoyed by schematic spaces.