I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S'$ is its generic point and Zariski points whose residue fields are finite are dense in $S'$.

More generally, for a scheme $S$ I wonder when there exists a smooth morphism (or an open cover) $S''\to S$ such that $S''$ is a pro-open pro-subscheme of $S'$ as above.

Are these properties of fields and schemes related to any existing terminology? Did anybody related anything similar to Deligne's weights on etale cohomology (and so, to Frobenius elements in absolute Galois groups)? Possibly, my assumptions on $S'$ can be modified. Any hints would be very welcome!

I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic point and Zariski points whose residue fields are finite are dense in $S'$.

More generally, for a scheme $S$ I wonder when there exists a smooth morphism (or an open cover) $S''\to S$ such that $S''$ is a pro-open pro-subscheme of $S'$ as above.

Are these properties of fields and schemes related to any existing terminology? Did anybody related anything similar to Deligne's weights on etale cohomology (and so, to Frobenius elements in absolute Galois groups)? Possibly, my assumptions on $S'$ can be modified. Any hints would be very welcome!