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We prove, for quasicompact separated schemes over ground fields, that Cechcohomology coincides with sheaf cohomology with respect to the Nisnevichtopology. This is a partial generalization of Artin's result that fornoetherian schemes such an equality holds with respect to the etale topologyunder the assumption that every finite subset admits an affine openneighborhood (AF-property). Our key result is that on the absolute integralclosure of separated algebraic schemes, the intersection of any two irreducibleclosed subsets remains irreducible. We prove this by establishing generalmodification and contraction results adapted to inverse limits of schemes.Along the way, we characterize schemes that are acyclic with respect to variousGrothendieck topologies, study schemes all whose local rings are strictlyhenselian, and analyze fiber products of strict localizations.

Geometry on totally separably closed schemes