  We show that on integral normal separated schemes whose function field is
separably closed, for each pair of points the intersection of the resulting
local schemes is local. This extends a result of Artin from rings to schemes.
The argument relies on the existence of certain modifications in inverse
limits. As an application, we show that \v{C}ech cohomology coincides with
sheaf cohomology for the Nisnevich topology. Along the way, we generalize the
characterization of contractible curves on surfaces by negative-definiteness of
the intersection matrix to higher dimensions, using bigness of invertible
sheaves on non-reduced schemes.
