Timeline for Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free?
Current License: CC BY-SA 2.5
21 events
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| Jun 29, 2010 at 15:29 | comment | added | Graham Leuschke | Congratulations Long! | |
| Jun 29, 2010 at 14:09 | comment | added | Hailong Dao | My son is about to come out, so see you guys in a few weeks (-: | |
| Jun 29, 2010 at 12:45 | comment | added | Hailong Dao | @Boyarsky: Ah, that's clear, but Bruns-Herzog also only assume flatness, it is in fact the same as EGA. I fixed the word now. Thanks for clarifying it. | |
| Jun 29, 2010 at 12:42 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Jun 29, 2010 at 10:35 | comment | added | Boyarsky | @Hailong: The EGA 6.3.4 result only assumes $N$ is finite over $S$, not over $R$, so it is a natural hypothesis; BCnrd's objection was about your hypothesis of finiteness over $R$. In other words, the EGA hypotheses are very well-behaved with respect to localization on top, whereas your earlier finiteness hypotheses over the base ring were not. Likewise, you are still insisting on freeness rather than just flatness for $N$ over $R$, and there as well your hypothesis is ruined by localizing on $S$; EGA only requires $R$-flatness, which is preserved by such localization. Think about it. | |
| Jun 29, 2010 at 3:57 | comment | added | Hailong Dao | BCnrd: I just looked at EGA IV, and 6.3.4 is precisely 1.2.16 of Bruns-Herzog, and it (or rather, the assumption of 6.3.1) assumes N finite. So I agree with you, one can not just quote EGA. | |
| Jun 29, 2010 at 3:47 | comment | added | BCnrd | Hailong, the combined reference "Auslander-Buchsbaum" (either the usual form, or just for regular rings as in the EGA ref. above) and the depth result in the EGA ref. above yield the conclusion, but really one does still need to make some arguments, especially tracking non-vanishing of stalks. So I sincerely hope Ben will state the general result with a finite map from regular to CM scheme and use refs to give a short argument to prove the result. Even with more regularity, if he prefers, the reader deserves a (short?) argument to go from whatever refs to the conclusion he wants. | |
| Jun 29, 2010 at 3:34 | comment | added | Hailong Dao | @BCnrd: perhaps you can post the EGA reference as an answer, as the OP really wanted a reference? | |
| Jun 29, 2010 at 3:29 | comment | added | Hailong Dao | I see what you mean now, it was not about the finiteness of the map! Thanks for all the comments. | |
| Jun 29, 2010 at 3:22 | comment | added | BCnrd | Hailong, I still object to your imposition of finiteness condition on the module when the ring is local upstairs. It is more natural to impose "semi-local" (and dimension conditions), since when you localize on top without completing then you will lose the $R$-finite/freeness. As for EGA being useless, I stand corrected: IV$_2$, 6.3.4 gives the required result under precisely the CM and regularity hypotheses you suggest, and applies under the semi-local hypotheses which I keep urging (since it only requires finiteness on fibers, which is the "right" condition). Grothedieck wins again! | |
| Jun 29, 2010 at 2:59 | comment | added | Hailong Dao | I am surprised EGA does not have this! | |
| Jun 29, 2010 at 2:56 | comment | added | Hailong Dao | That's right, I never assume flatness. Anyway, I checked the references in Bruns-Herzog more carefully and I don't think we need finiteness, but need same dimension. I edited my answer, hope it is OK now. | |
| Jun 29, 2010 at 2:54 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Jun 29, 2010 at 2:42 | comment | added | BCnrd | Hailong, assuming regular upstairs and CM downstairs, the proof in the accepted answer works without any need for flatness! As for relevant EGA references, not much to say. They prove Auslander-Buchsbaum only in the regular case (all we need) in $0_ {\rm{IV}}$, 17.3.4 (so a rare case when EGA missed optimal generality; for shame), and then 17.3.5(i) says the obvious conclusion that local freeness on top amounts to have full depth on stalks. But to verify that condition, one has to go through the argument in the accepted answer, adapted to the weaker hypotheses; so EGA is useless (for this). | |
| Jun 29, 2010 at 2:19 | history | edited | BCnrd | CC BY-SA 2.5 |
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| Jun 29, 2010 at 1:20 | comment | added | BCnrd | Hailong, I am surprised the CM reference does not require flatness. Anyway, if you just localize at top then you lose the finiteness hypothesis; that's why I asked about weakening to semi-local (and it is the reason completions -- or henselizations -- would have to intervene, as in the accepted answer, to bust apart semi-local into a direct product of locals). As for EGA ref., let me get back to you about that... | |
| Jun 28, 2010 at 23:24 | comment | added | Hailong Dao | @BCnrd: Also, may be EGA has some better reference? If you know, please share with us. Thanks. | |
| Jun 28, 2010 at 23:02 | comment | added | Hailong Dao | BCnrd: The reference does not require $S$ to be $R$-flat. If $S$ is semi-local, can't we just localize at each maximal ideal of $S$? | |
| Jun 28, 2010 at 22:49 | comment | added | BCnrd | Hailong, in (1) it is restrictive to demand finiteness while also requiring $S$ to be local (rather than just semi-local). For example, in the chosen answer an extra argument with completions is needed to pass to the "finite local" case from the "finite semi-local" case. So for the CM case (in which case the trick of "connected=irreducible" is less relevant), it is better (for purpose of "global" applications) to take $S$ just semi-local and then need more argument to get to the local case. Does your CM reference apply verbatim for $R$-flat (?) semi-local $S$? | |
| Jun 28, 2010 at 22:48 | history | edited | Hailong Dao | CC BY-SA 2.5 |
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| Jun 28, 2010 at 20:06 | history | answered | Hailong Dao | CC BY-SA 2.5 |