Timeline for Can "ampleness" be detected inside the derived category?
Current License: CC BY-SA 3.0
19 events
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| Dec 19, 2022 at 8:34 | vote | accept | Saal Hardali | ||
| Dec 19, 2022 at 17:50 | |||||
| Dec 15, 2017 at 13:17 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| S Nov 23, 2017 at 13:39 | history | bounty ended | CommunityBot | ||
| S Nov 23, 2017 at 13:39 | history | notice removed | CommunityBot | ||
| Nov 16, 2017 at 19:10 | comment | added | Jeremy Rickard | Also, you seem to suggest that knowing the $\otimes$-structure implies knowing the $\text{Pic}(X)$-action. Is this clear? It would be if the $\otimes$-structure determined the line bundles, rather than just their shifts. | |
| Nov 16, 2017 at 19:07 | comment | added | Jeremy Rickard | For any line bundle $\mathcal{L}$, the functor $-\otimes\mathcal{L}$ is a self-equivalence of the derived category that commutes with the action of $\text{Pic}(X)$, so I don't think you can distinguish different line bundles without using the $\otimes$-structure, or only using the $\text{Pic}(X)$-action. | |
| Nov 15, 2017 at 12:23 | history | edited | Saal Hardali | CC BY-SA 3.0 |
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| S Nov 15, 2017 at 12:20 | history | bounty started | Saal Hardali | ||
| S Nov 15, 2017 at 12:20 | history | notice added | Saal Hardali | Draw attention | |
| Nov 13, 2017 at 8:01 | history | edited | Saal Hardali | CC BY-SA 3.0 |
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| Nov 13, 2017 at 7:59 | comment | added | Saal Hardali | @YosemiteSam Edited so that the question would still be interesting despite Tannakian wizardry. | |
| Nov 13, 2017 at 7:57 | comment | added | Saal Hardali | @მამუკაჯიბლაძე Of course I agree, still the fact that these two classical definitions agree has always been a miracle for me. I can't think of a proof which doesn't reduce to some form of computation (e.g. serre vanishing). | |
| Nov 13, 2017 at 7:55 | history | edited | Saal Hardali | CC BY-SA 3.0 |
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| Nov 13, 2017 at 7:45 | history | edited | Saal Hardali | CC BY-SA 3.0 |
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| Nov 13, 2017 at 5:32 | comment | added | მამუკა ჯიბლაძე | Well I would say "mysterious" depends on point of view. From another (maybe old-fashioned) point of view, describing all possible maps to the projective space is what the line bundles are for. | |
| Nov 13, 2017 at 3:06 | comment | added | Yosemite Sam | he means DQCoh(X), or maybe some infinity version of that. The question can be answered by cheating, as the tensor structure allows you to recover X itself (and therefore any ample line bundle on it). But I don't know how to define ampleness intrinsically. | |
| Nov 13, 2017 at 1:00 | comment | added | R. van Dobben de Bruyn | Usually, $\operatorname{QCoh}(X)$ refers to the category of quasi-coherent sheaves on $X$, not its associated derived category. Which of the two are you interested in? | |
| Nov 12, 2017 at 22:59 | answer | added | Leo Alonso | timeline score: 5 | |
| Nov 12, 2017 at 22:49 | history | asked | Saal Hardali | CC BY-SA 3.0 |