Geometry on totally separably closed schemes

Abstract

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.