2021/05/05 by J. Sánchez Gonzalez, J. Sánchez González, Carlos Prieto +3
Mathematics · #06A11 #14A15 #14E20 #18E50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.AT #msc:06A11 #msc:14A15 #msc:14E20 #msc:18E50
paper · pdf · doi:10.48550/arxiv.2105.01947
arxiv created 2021/05/05 · openalex publication_date 2021/05/05 · arxiv updated 2021/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the category of finite étale 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 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. 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.