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.
Keywords
Absolute algebraic closure, acyclic schemes, étale and
Nisnevich topology, henselian rings, Čech and sheaf
cohomology, contractions
