Timeline for Finite, Étale Morphism Of Varieties
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 9, 2020 at 14:52 | comment | added | Lao-tzu | @Anton Geraschenko I think I can prove it now using degree formula. | |
| Apr 9, 2020 at 14:36 | comment | added | Lao-tzu | @Anton Geraschenko Is the finite etale map $(Y\times_X Y -\Delta)\to Y$ surjective (if $deg(\pi)>1$)? I think you need this to see that the $U$ you get in the end turns out to be an etale cover of $X$. I'm unable to show this. | |
| Jan 2, 2013 at 14:58 | comment | added | Anton Geraschenko | @ZhuangXiaobo: Let's work locally, so that the map is $Spec(S)\to Spec(R)$. By definition, "degree $n$" means that the corresponding ring homomorphism $R\to S$ makes $S$ into a free module of rank $n$ over $R$. If $n=1$, the map is bijective, so an isomorphism of rings. | |
| Jan 2, 2013 at 2:30 | comment | added | Xiaobo Zhuang | @Anton Geraschenko:How to see an finite etale morphism of degree one is an isomorphism? I think you use this fact in your answer. | |
| Aug 8, 2011 at 17:44 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| Aug 8, 2011 at 16:38 | comment | added | Anton Geraschenko | It's worth pointing out that it's a general philosophy that things that you expect to be true because of your experience with the analytic topology are usually true in the etale topology. | |
| Aug 8, 2011 at 16:26 | comment | added | Akhil Mathew | Another way to see the last fact is to note that finite etale covers of a strictly henselian ring are split, together with the fact that the strict henselianization at a point is the "limit" of etale neighborhoods (and a "noetherian descent" argument to argue that the splitting must descend to some etale neighborhood after all). | |
| Aug 8, 2011 at 16:07 | vote | accept | Jesko Hüttenhain | ||
| Aug 8, 2011 at 16:00 | history | answered | Anton Geraschenko | CC BY-SA 3.0 |