Timeline for Surjective implies local affine surjective?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 14, 2011 at 12:33 | vote | accept | evgeniamerkulova | ||
| Feb 13, 2011 at 21:34 | answer | added | Qing Liu | timeline score: 7 | |
| Feb 13, 2011 at 21:31 | comment | added | Ariyan Javanpeykar | Ok so you're not (really) asking about affine morphisms because U doesn't need to be the inverse image of V. Again apologies. Thank you Qing Liu. | |
| Feb 13, 2011 at 21:29 | comment | added | Ariyan Javanpeykar | Oops. Sorry. I guess I should have looked better. | |
| Feb 13, 2011 at 21:23 | comment | added | Ariyan Javanpeykar | To see that you are asking about "affine morphisms" you could look at chap. II, exercise 5.17 of Hartshorne's Algebraic geometry. (Not completely sure about the exercise number because I don't have access to the book right now.) | |
| Feb 13, 2011 at 21:23 | comment | added | Qing Liu | @Ariyan: if $Y$ is one point, any any non-empty affine open subset of $X$ maps surjectively to $Y$. So $f$ always satisfies the required property in this case. | |
| Feb 13, 2011 at 21:12 | comment | added | Ariyan Javanpeykar | Take Y=Spec k, with k a field. Let f:X---> Y be a morphism. (It is automatically surjective.) Then f satisfies your condition if and only if X is affine. What you are asking for is the "relativization" of this. That is, your morphism f will satisfy your condition if it is "affine". Let me emphasize here that the morphism needs to be affine (and not the schemes necessarily.) Finite morphisms are affine. When they are surjective they are called (branched) covers. It is not true that every surjective affine morphism is finite. Consider for example the projection of A^n to A^1. | |
| Feb 13, 2011 at 19:52 | history | edited | evgeniamerkulova | CC BY-SA 2.5 |
I ask if question become true with more hypotheses
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| Feb 13, 2011 at 19:16 | answer | added | Kevin Ventullo | timeline score: 6 | |
| Feb 13, 2011 at 18:25 | history | asked | evgeniamerkulova | CC BY-SA 2.5 |