Geometry on totally separably closed schemes

TL;DR

This paper demonstrates that for certain schemes, Cech and sheaf cohomology agree in the Nisnevich topology, and it establishes irreducibility of intersections in the absolute integral closure, advancing understanding of scheme cohomology and structure.

Contribution

It generalizes Artin's result to a broader class of schemes and introduces new methods for analyzing inverse limits and irreducibility in scheme theory.

Findings

Cech cohomology coincides with sheaf cohomology in the Nisnevich topology for certain schemes.

Intersections of irreducible closed subsets in the absolute integral closure remain irreducible.

Characterization of schemes acyclic in various Grothendieck topologies.

Abstract

We prove, for quasicompact separated schemes over ground fields, that Cech 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 etale topology 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 whose local rings are strictly henselian, and analyze fiber products of strict localizations.