Timeline for Making a quasi-compact open into an affine open
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jun 18, 2019 at 13:04 | history | bounty ended | CommunityBot | ||
| S Jun 18, 2019 at 13:04 | history | notice removed | CommunityBot | ||
| S Jun 10, 2019 at 11:25 | history | bounty started | CommunityBot | ||
| S Jun 10, 2019 at 11:25 | history | notice added | user141498 | Authoritative reference needed | |
| May 18, 2019 at 5:18 | comment | added | user138661 | @HarryGindi what is not true then? Could you kindly give an explicit example where there is no scheme structure on $X$ (affine or non-affine), whose restriction to $U$ would define an affine scheme structure? | |
| May 17, 2019 at 21:50 | comment | added | Harry Gindi | Oh, I misread the question. No, it's not true then. You can equip U with the restriction of the structure sheaf to be non-affine, then take the ring of global sections and U will be homeomorphic to the spec of the ring of global sections. That is, $U\cong \operatorname{Spec}(\mathcal{O}_X(U))$ as topological spaces (not as schemes!!!) This is because the underlying space of any qcqs scheme is homeo to spec of the ring of global sections. | |
| May 17, 2019 at 17:26 | comment | added | user138661 | @HarryGindi I do not understand this. Sure, you can put an affine scheme structure on $U$. Why would it extend to $X$? If I am missing something obvious, well, sorry, I am bad with this stuff. | |
| May 17, 2019 at 17:08 | comment | added | Harry Gindi | Yes, it is actually separated and qc, so this follows by Hochster's characterization as the qcqs sober spaces with topology generated by qc opens. Separation follows from: stacks.math.columbia.edu/tag/01P5 | |
| May 17, 2019 at 13:00 | comment | added | Harry Gindi | I think that this question is equivalent then to asking if a qc open subspace of a spectral space is quasiseparated by Hochster's theorem. I think it's true, but I'd need to work it out. | |
| May 17, 2019 at 12:37 | history | asked | user138661 | CC BY-SA 4.0 |