Timeline for Blowing up a derived scheme
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2013 at 20:39 | answer | added | guest | timeline score: 0 | |
| Nov 29, 2011 at 20:22 | comment | added | Moosbrugger | With that said, I don't see any problem formulating the universal property for blow-ups in the derived setting -- the notion of Cartier divisor is no problem. However, the usual proof of representability doesn't go through, and it may take some fiddling with deformation theory to show that it is representable. | |
| Nov 29, 2011 at 20:19 | comment | added | Moosbrugger | I'm not sure what to say -- it's hard to argue why a notion does not exist. If you look in Sections 4 and 5 of DAG XII, you can see Lurie avoiding the notion of $n$th power of an ideal, but I don't see anywhere that he explicitly says that it doesn't exist or why. The essential notion which is missing here is image of an ideal under a map, which doesn't exist since image isn't a notion in a stable $\infty$-category. By the way, even the notion of ideal is fraught in the derived setting: e.g., the only invariant notion of quotient I know of is a map $A\to B$ which is surjective on $\pi_0$. | |
| Nov 29, 2011 at 5:09 | comment | added | thel | Moosbrugger: this is exactly what I feared. Maybe we can entice somebody to post an example of what is going wrong? I'm planning to complete my rings at the end of the day anyway, so a little hope remains. | |
| Nov 29, 2011 at 3:48 | comment | added | Moosbrugger | I've heard from experts that the answer is no: there is not a good notion of powers of an ideal. (Say, invariant under quasi-isomorphism, or definable in Lurie's framework). So there's no $n$th infinitesimal neighborhoods in DAG, only formal completions. | |
| Nov 29, 2011 at 3:41 | answer | added | Sean Sather-Wagstaff | timeline score: 4 | |
| Nov 24, 2011 at 2:55 | history | asked | thel | CC BY-SA 3.0 |