Timeline for Recover a morphism from its pullback
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 6, 2010 at 15:58 | vote | accept | Martin Brandenburg | ||
| Sep 7, 2010 at 16:19 | |||||
| Sep 1, 2010 at 9:36 | comment | added | Laurent Moret-Bailly | No, I agree. So my "answer" is really a comment. | |
| Aug 31, 2010 at 17:38 | comment | added | Martin Brandenburg | See my edit in the question; I think we need more than $f=g$ set-theoretically in order to reduce to the affine case. Or do I miss something? | |
| Aug 31, 2010 at 11:03 | vote | accept | Martin Brandenburg | ||
| Aug 31, 2010 at 11:26 | |||||
| Aug 31, 2010 at 9:51 | comment | added | Martin Brandenburg | Ah I see: $(f^* i_* \mathcal{O}_Z)_x=0$ if and only if $\mathcal{O}_{X,x}$ is generated by $f^\#(I_{f(x)})$, if and only if $I_{f(x)} = \mathcal{O}_{Y,f(x)}$ since $f^\#$ is a local homomorphism! Thus we don't have to know $f^\#$. | |
| Aug 31, 2010 at 9:31 | comment | added | Laurent Moret-Bailly | Yes. Of course the $\mathcal{O}_X$-module structure of this thing is not enough to get the surjection $\mathcal{O}_X\to\mathcal{O}_Z$ but it does determine the support. | |
| Aug 31, 2010 at 9:30 | comment | added | Martin Brandenburg | I see that $f^{-1}(Z)$ is the support of $f^{-1} i_* \mathcal{O}_Z$, but $f^* i_* \mathcal{O}_Z$ could have a smaller support and this depends on $f$. | |
| Aug 31, 2010 at 9:24 | comment | added | Martin Brandenburg | Ah this is the argument I was lookin' for. You mean the pushforward of $\mathcal{O}_Z$ with $Z \to X$, right? | |
| Aug 31, 2010 at 9:08 | history | answered | Laurent Moret-Bailly | CC BY-SA 2.5 |