  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.
