2019/11/05 by Marcin Lara, Lara, Marcin
Mathematics · #13B40 #13J15 #14F20 #14F35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT #msc:13B40 #msc:13J15 #msc:14F20 #msc:14F35
paper · pdf · doi:10.48550/arxiv.1911.01884
21 pages, comments welcome!
arxiv created 2019/11/05 · openalex publication_date 2019/11/05 · arxiv updated 2019/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual étale fundamental group π1et defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite étale covers. We prove exactness of the general homotopy sequence for the pro-étale fundamental group, i.e. that for a geometric point s on S and a flat proper morphism X → S of finite presentation whose geometric fibres are connected and reduced, the sequence π1proet(X_s) → π1proet(X) → π1proet(S) → 1 is "nearly exact". This generalizes a theorem of Grothendieck from finite étale covers to geometric coverings. We achieve the proof by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization in this setting.