2015/03/31 by Stefan Schröer
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cohomology #Combinatorics #Discrete mathematics #Equivariant cohomology #Homotopy and Cohomology in Algebraic Topology #Mathematics #Noetherian #Pure mathematics #Sheaf #Topology (electrical circuits) #math.AG #msc:13B22 #msc:13J15 #msc:14E05 #msc:14F20 #Étale cohomology
paper · pdf · doi:10.2140/ant.2017.11.537
published as Alg. Number Th. 11 (2017) 537-582 · 37 pages, minor changes, references added, appendix on inductive dimension included, to appear in Algebra Number Theory
arxiv created 2016/11/21 · openalex publication_date 2017/05/06 · arxiv updated 2017/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove, for quasicompact separated schemes over ground fields, that Čech cohomology coincides with sheaf cohomology with respect to the Nisnevich topology. This is a partial generalization of Artin’s result that for noetherian schemes such an equality holds with respect to the étale topology, which holds under the assumption that every finite subset admits an affine open neighborhood (AF-property). Our key result is that on the absolute integral closure of separated algebraic schemes, the intersection of any two irreducible closed subsets remains irreducible. We prove this by establishing general modification and contraction results adapted to inverse limits of schemes. Along the way, we characterize schemes that are acyclic with respect to various Grothendieck topologies, study schemes all local rings of which are strictly henselian, and analyze fiber products of strict localizations.