Timeline for Does Qcoh(X) admit a generating set?
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22 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 2, 2010 at 19:11 | vote | accept | Martin Brandenburg | ||
| Oct 2, 2010 at 18:27 | answer | added | Thomas Nevins | timeline score: 9 | |
| Sep 29, 2010 at 19:01 | answer | added | Martin Brandenburg | timeline score: 12 | |
| Sep 29, 2010 at 17:39 | comment | added | Thomas Nevins | Gabber's argument also appears in print in Enochs and Estrada, "Relative homological algebra in the category of quasi-coherent sheaves," Adv. in Math. 194 (2005) 284--295. | |
| Sep 29, 2010 at 0:48 | comment | added | BCnrd | Martin, good to hear! I probably have the original in a filing cabinet somewhere, but am not sure. If you do get around to typing it up, please email me the .pdf file. Thanks. | |
| Sep 28, 2010 at 14:36 | comment | added | Martin Brandenburg | Gabber has sent me a scan of a 11 year old letter, which is addressed to BCnrd ;). I will try to write it up. | |
| Sep 27, 2010 at 14:30 | comment | added | Philipp Hartwig | @Bugs Bunny: The first sentence of my comment refers to an earlier version of the question which mentioned the abelian category QCoh(X). | |
| Sep 27, 2010 at 13:13 | comment | added | Martin Brandenburg | It's not local. Check the details and you will run into the problems which are solved in the case of concentrated schemes. | |
| Sep 27, 2010 at 12:49 | comment | added | Bugs Bunny | @ M.B. Of course you know, this means war! :-) Me thinking that all you need is to distinguish ($af\neq bf$) a pair of distinct morphisms $a,b:X->Y$ by mapping generator to $f:?->X$. Since $a$ and $b$ are distinct, they are distinct on some stalk $X_t$. So all we need is a generator for each $QCoh_t$, of sheaves supported at $t$. But this seems to be a local question, so we are just talking about modules over a local ring, so some appropriate skyscraper is a generator... Have I totally gone mad? | |
| Sep 27, 2010 at 11:53 | comment | added | Martin Brandenburg | @Bugs: Homomorphisms on skyscapers only give global sections. | |
| Sep 27, 2010 at 11:05 | comment | added | Bugs Bunny | @ Martin B. Doc, you may think it is bonkers but why don't skyscrapers at all the point form a generating set??? | |
| Sep 27, 2010 at 11:00 | comment | added | Bugs Bunny | @ Phillip H. Non-abelian categories may still have generators. One does not have much to do with another... | |
| Sep 26, 2010 at 18:25 | comment | added | Martin Brandenburg | I didn't mean all the details. Anyway, I have emailed Gabber. | |
| Sep 26, 2010 at 15:57 | comment | added | BCnrd | Dear Martin: I don't remember how Gabber's argument goes (except that it was "soft", not needing anything deep, just some set-theoretic cleverness -- I think $\kappa$ involves the cardinality of some open affine cover and open affine covers of the double overlaps thereof, and probably other stuff too such as coordinate rings of affine opens), and I don't have time to reconstruct it. I hope someone who is really into such matters may do that, and then they can post it as an appropriate answer. That will be better. | |
| Sep 26, 2010 at 9:46 | comment | added | Martin Brandenburg | @Brian: Great. Could you post this as an answer? | |
| Sep 25, 2010 at 14:33 | comment | added | BCnrd | There are no counterexamples. For an arbitrary scheme $X$ there exists an infinite cardinal $\kappa$ so that every quasi-coherent sheaf is generated by its quasi-coherent subsheaves of type $\kappa$, where the latter means that sections over some open affine cover (and then necessarily over any open affine) are generated by $\le \kappa$ elements as a module. This was explained to me long ago by Gabber, so ask him for the details. There is obviously a set of isomorphism class representatives for the quasi-coherent sheaves of type $\kappa$, so that settles it affirmatively in general. | |
| Sep 25, 2010 at 14:32 | answer | added | D.-C. Cisinski | timeline score: 12 | |
| Sep 25, 2010 at 12:43 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
added 211 characters in body
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| Sep 25, 2010 at 12:37 | comment | added | Martin Brandenburg | Thank you Philipp ;). The notes of Daniel Murfet are very good (and remind me of the stacks project). | |
| Sep 25, 2010 at 11:43 | comment | added | Philipp Hartwig | For a ringed space X the category QCoh(X) need not be abelian. The usual hypothesis on a scheme X which ensures that QCoh(X) is a Grothendieck category is that X be quasi-compact and quasi-separated (this is often called "concentrated"). See [Lipman-Notes on derived functors and Grothendieck duality, 4.1.3.1] (available on his website) for a proof in this case. Also Daniel Murfet's notes are useful and contain a proof, see [therisingsea.org/notes/ModulesOverAScheme.pdf, Proposition 66]. I haven't thought about counterexamples in other cases. | |
| Sep 25, 2010 at 10:47 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |