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Suppose that a scheme $S$ is separated, excellent, and has finite Krull dimension. Which of the following statements are true:

  1. If $S$ is regular, then it can be presented as a projective limit of smooth $\mathbb{Z}$-schemes.

  2. If $S$ is regular, then it can be presented as a projective limit of schemes that are smooth over finite type regular $\mathbb{Z}$-ones.

  3. Any (i.e. not necessarily regular) $S$ can be presented as a projective limit of schemes that are smooth over finite type $\mathbb{Z}$-ones.

Also, could one assume the connecting morphisms in the limits above to be affine?

I would be deeply grateful for any statements, examples, or references here! Are there any additional restrictions on $S$ needed? Is such a statement true at least locally? Are there any counter-examples known?

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  • $\begingroup$ It would be nice if you motivate these questions (where are they helpful)? $\endgroup$
    – Martin Brandenburg
    Commented Feb 7, 2011 at 16:53
  • $\begingroup$ Well, I want to use some weights for $S$-sheaves and motives. If I am able to present $S$ as a limit of $S_i$, $S_i$ is smooth over $T_i$ and $T_i$ is a finite type $\mathbb{Z}[1/l]$-scheme, then it would be sufficient to relate weights and motives over all possible $T_i$. Actually, it would be sufficient for me to have an open covering of $S$ that satisfies this property. $\endgroup$
    – Mikhail Bondarko
    Commented Feb 7, 2011 at 17:03
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    $\begingroup$ I do not know whether this is something of interest. Dorin Popescu proved that any regular local ring (say containing a field) is the direct limit of essentially finite type (over a field) regular local rings. The proof is difficult and the result very useful. I am sure you can find the references on the web. Swan later wrote up Popescu's proof with more details. I do not recall what he proves for schemes over the integers. $\endgroup$
    – Mohan
    Commented Feb 7, 2011 at 23:10
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    $\begingroup$ If you only need it for an open covering, then you can assume that $S$ is affine. In that case, as Mohan points out, Popescu's theorem will give the answer to your first question. It says that a Noetherian ring is regular if and only if it is a direct limit of smooth rings. (Now the result is know even for rings that don't contain a field.) For the statement of Popescu's theorem and a discussion of proofs (with references), you can look at the following article of B. Conrad and de Jong : math.stanford.edu/~conrad/papers/approx.pdf (theorem 1.3 and remark 1.4). $\endgroup$
    – Alex
    Commented Feb 8, 2011 at 2:19

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