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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!

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  • $\begingroup$ Clearly, the condition is satisfied if $K$ is finitely generated (over the prime field). Do you know other examples? $\endgroup$
    – Laurent Moret-Bailly
    Commented Oct 27, 2019 at 8:05
  • $\begingroup$ Possibly, you are right, and one cannot get anything else under my assumptions.:) I will think about weakening them. $\endgroup$
    – Mikhail Bondarko
    Commented Oct 27, 2019 at 9:23

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