Timeline for A presentation of a scheme as a limit of smooth ones over finitely generated bases
Current License: CC BY-SA 2.5
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| Feb 8, 2011 at 2:19 | comment | added | Alex | 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). | |
| Feb 7, 2011 at 23:10 | comment | added | Mohan | 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. | |
| Feb 7, 2011 at 20:03 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
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| Feb 7, 2011 at 17:03 | comment | added | Mikhail Bondarko | 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. | |
| Feb 7, 2011 at 16:53 | comment | added | Martin Brandenburg | It would be nice if you motivate these questions (where are they helpful)? | |
| Feb 7, 2011 at 12:35 | history | edited | Mikhail Bondarko | CC BY-SA 2.5 |
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| Feb 7, 2011 at 12:27 | history | asked | Mikhail Bondarko | CC BY-SA 2.5 |