Timeline for Existence and uniqueness of extensions of a finite flat map
Current License: CC BY-SA 3.0
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 22, 2015 at 17:49 | comment | added | Nicholas Proudfoot | I'm still confused about a technical point. Assume for simplicity that $f_*{\scr{O}}_X$ is trivial. Choose a trivialization, and use your procedure to define an extension $X'$ in codimension 1. Is it clear that $f'_*{\scr{O}}_{X'}$ is still trivial? If not, then I'm worried that perhaps $f'_*{\scr{O}}_{X'}$ does not extend to a vector bundle on $S$. | |
| May 5, 2015 at 20:28 | comment | added | David E Speyer | @NicholasProudfoot Yes, your second comment is the right one. Thanks for thinking it through; you said that better than I would have. | |
| May 5, 2015 at 17:16 | comment | added | Nicholas Proudfoot | No, I guess that what I wrote above is wrong. If I understand correctly, the difference between the two examples in Tyler's comment is the choice of extension of $f_*\mathcal{O}_X$. Such an extension is a subtle object--it is a locally free sheaf $\mathcal{E}$ on $S$ along with an isomorphism $\mathcal{E}|_U\cong f_*\mathcal{O}_X$. In Tyler's two examples, $X$ and $X'$ are isomorphic, as are the implied extensions $\mathcal{E}$ and $\mathcal{E}'$, but the isomorphism between $\mathcal{E}$ and $\mathcal{E}'$ cannot be chosen to be compatible with the isomorphism between $X$ and $X'$. | |
| May 4, 2015 at 18:29 | comment | added | Nicholas Proudfoot | Just to be clear, is it correct that the Hilbert scheme that you refer to is non-separated, and this is to blame for the non-uniqueness in Tyler Lawson's example? | |
| May 1, 2015 at 11:03 | comment | added | David E Speyer | Right. And there always is an extension in codimension $\leq 2$. At least, that is what I am picking up from the conversation in mathoverflow.net/questions/120776 and mathoverflow.net/questions/84597 and section 4 of ams.org/mathscinet-getitem?mr=169877 , although I haven't found an official place that says so. | |
| May 1, 2015 at 5:11 | vote | accept | Nicholas Proudfoot | ||
| May 1, 2015 at 5:10 | comment | added | Nicholas Proudfoot | Oh, sorry, I guess that your comment above shows that existence can fail if and only if there is a bundle on $U$ that doesn't extend to $S$. Thanks! | |
| May 1, 2015 at 5:06 | comment | added | Nicholas Proudfoot | Hi David, thanks for the detailed answer! One question: You give a somewhat complicated counterexample to existence of extensions in which $S\smallsetminus U$ has codimension 3. Is it possible that existence holds when the codimension is equal to 1? I guess that Tyler Lawson's example shows that uniqueness can fail when the codimension is 1. | |
| May 1, 2015 at 2:49 | history | edited | David E Speyer | CC BY-SA 3.0 |
added 1025 characters in body
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| May 1, 2015 at 2:28 | history | answered | David E Speyer | CC BY-SA 3.0 |