Timeline for Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?
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13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 19, 2011 at 12:06 | comment | added | Mikhail Bondarko | By the 'topological situation' I meant lifting open subsets from closed submanifolds. Yet note that a tubular neighbourhood of $X$ in $Y$ has the same fundamental group as $X$, so one definitely does not obtain a counter-example this way. | |
| Sep 18, 2011 at 15:46 | comment | added | David E Speyer | It's worth pointing out that the topological version of this is also untrue, with the same counterexample and reason. | |
| Sep 18, 2011 at 12:50 | comment | added | Donu Arapura | OK, I stand corrected. | |
| Sep 18, 2011 at 8:00 | comment | added | Laurent Moret-Bailly | Perhaps you can add SGA2, exposé X to the references. | |
| Sep 18, 2011 at 6:12 | comment | added | Mikhail Bondarko | Thanks for the comments!! I probably need etale tubular neighbourhoods anyway (though it would be interesting to understand all alternatives here); I will also look at the references mentioned. | |
| Sep 18, 2011 at 6:09 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
added 129 characters in body; edited title
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| Sep 18, 2011 at 5:06 | comment | added | Torsten Ekedahl | @Donu: Note that the extension is not required to be a finite étale map. | |
| Sep 18, 2011 at 2:55 | comment | added | Jonathan Wise | Such a statement is true Zariski locally. See EGA IV, Proposition 18.1.1. | |
| Sep 17, 2011 at 21:57 | comment | added | Donu Arapura | There is also some old work of Cox in the simplicial scheme setting, which maybe worth looking at as well. | |
| Sep 17, 2011 at 21:56 | comment | added | Donu Arapura | Mikhail, I'll assume the correction suggested by Matthieu, but then it isn't true. Take $Y=\mathbb{P}^2$ and $X\subset Y$ a smooth elliptic curve. Any etale cover induced from $Y$ is trivial, but the fundmental group of $X$ is large. You probably want some sort of tubular neighbourhood, but getting this in algebraic geometry seems tricky. If you know Marc Levine, you might ask him. I think he has some version of this, but I haven't seriously looked at it. | |
| Sep 17, 2011 at 20:57 | comment | added | Matthieu Romagny | You probably mean "let $U/X$ be étale". | |
| Sep 17, 2011 at 20:41 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |