Timeline for Embedding dimension=minimum dimension of a local embedding?
Current License: CC BY-SA 3.0
9 events
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| May 19, 2012 at 19:04 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| May 19, 2012 at 15:02 | comment | added | Will Sawin | I hope that is more clear. I will edit the answer to make it more clear when I have time. | |
| May 19, 2012 at 15:02 | comment | added | Will Sawin | 1. Taking the smallest closed subscheme it factors through is fine, because that satisfies all the relations between $x_1,...,x_m$ that hold in the local ring. 2. Yes, I mean the formal completion. 3. The map is the vanishing set of a set of #n# equations in $n$-space. This is a characterization of etale maps, from Milne. 4. The cover comes from lifting the equations whose vanishing set is locally $U$ to $k[x_1,...,x_m]$ from $k[x_1,..,x_m]/I$, this produces a cover of $\textrm{Spec}k{x_1,...,x_m]$. Restricting the cover to $V(I)$ gives $U$, so $U$ factors through the cover. | |
| May 19, 2012 at 5:23 | comment | added | A. Pascal | Upon more careful reading, there are some things I don't understand. First, do you mean "has image some closed subscheme" or rather "factors through a closed subscheme"? Also, I presume you mean formal completion instead of profinite completion. Given that, we have the the map is etale at $P$. But then I don't know what "it is given by a set" means. "It" would seem to refer to the map, but I don't follow here. The map is already given, no? Finally, where does the cover come from and why does the map from $U$ lift to the cover? | |
| May 19, 2012 at 0:54 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| May 18, 2012 at 22:36 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| May 18, 2012 at 22:33 | comment | added | Will Sawin | I think everything I said works Zariski locally. Maybe I missed something though. | |
| May 18, 2012 at 16:42 | comment | added | A. Pascal | Thanks. So the statement I want is true etale locally. But do you think it is true Zariski locally (with a more complicated proof)? | |
| May 18, 2012 at 16:21 | history | answered | Will Sawin | CC BY-SA 3.0 |