Timeline for Can you give an example of two projective morphisms of schemes whose composition is not projective?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2019 at 21:15 | vote | accept | Georges Elencwajg | ||
| Nov 13, 2019 at 12:58 | history | edited | Georges Elencwajg | CC BY-SA 4.0 |
Cleared a terminological confusion
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| Nov 13, 2019 at 10:23 | comment | added | Georges Elencwajg | Thanks a lot for your comments, dear @ R. van Dobben de Bruyn. And I have modified the two possible conditions on $Z$ according to your first comment. | |
| Nov 13, 2019 at 10:20 | history | edited | Georges Elencwajg | CC BY-SA 4.0 |
Corrected conditions on $Z$ in accord with R. van Dobben de Bruyn comment
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| Nov 13, 2019 at 9:18 | history | edited | R. van Dobben de Bruyn |
edited tags
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| Nov 13, 2019 at 9:14 | answer | added | R. van Dobben de Bruyn | timeline score: 17 | |
| Nov 13, 2019 at 6:20 | comment | added | R. van Dobben de Bruyn | It seems to me that quasi-separatedness is only used in Tag 07RM to show that $(g \circ f)_* \mathscr L$ is a filtered colimit of finite type quasi-coherent $\mathcal O_Z$-modules. This also holds for example when $Z$ is locally Noetherian because the projective morphisms $f$ and $g$ preserve coherent sheaves under pushforward. In general however it could happen that $g \circ f$ is quasi-projective and proper, but not projective (but again you would need some reason why there isn't some other projective embedding). | |
| Nov 13, 2019 at 6:16 | comment | added | R. van Dobben de Bruyn | The most obvious (and probably most interesting) way for this to fail would be a non-quasi-compact $Z$ such that the power $a$ needed for $\mathscr L \otimes f^* \mathscr M^{\otimes a}$ to be $(g \circ f)$-ample (see Tag OC4K) is unbounded if you run over all affine opens in $Z$. But then you need a reason why you couldn't have chosen some other line bundle (so $Z$ a disjoint union of points is not going to work). | |
| Nov 13, 2019 at 4:50 | comment | added | R. van Dobben de Bruyn | Slight correction in the conditions: either $Z$ is quasi-compact and separated (stacks project: quasi-compact and quasi-separated), or $\operatorname{sp}(Z)$ is Noetherian. | |
| Nov 12, 2019 at 10:27 | history | asked | Georges Elencwajg | CC BY-SA 4.0 |