Timeline for Moduli stacks and representability of diagonal by schemes
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2021 at 21:30 | comment | added | Wojowu | @AriyanJavanpeykar I'm not able to fully follow this argument, but I find it believable. I don't mind the extra hypotheses you took - they are (at least in practice, I imagine) reasonably mild niceness properties. Feel free to post your argument as an answer. | |
| Mar 11, 2021 at 9:34 | comment | added | Ariyan Javanpeykar | ....I did add some additional hypotheses of finite presentation and separated diagonal. Were you trying to avoid these? | |
| Mar 11, 2021 at 9:33 | comment | added | Ariyan Javanpeykar | @Wojuwu Let $X\to S$ be a finitely presented DM (algebraic) stack (with separated diagonal). Let $\Delta$ be its diagonal. Let $T\to X\times_S X$ be a morphism with $T$ a scheme. Since $\Delta$ is unramified, this means that the pull-back of $\Delta$ along $T\to X\times_S X$ is an algebraic space, let's call it $I$, which is unramified over $T$. Now, since $I\to T$ is unramified separated and finitely presented, it is quasi-finite separated. An algebraic space which is quasi-finite separated over a scheme is a scheme. So, $I$ is a scheme, as required.... | |
| Mar 10, 2021 at 15:39 | comment | added | Wojowu | @AriyanJavanpeykar Apologies if this is a sign of my ignorance, but I'm not aware how to conclude from diagonal being unramified that it is representable by schemes. | |
| Mar 10, 2021 at 14:42 | comment | added | Ariyan Javanpeykar | I am probably misunderstanding Q2. But isn't the diagonal of a DM stack by definition unramified, and thus representable? | |
| Mar 8, 2021 at 23:09 | history | asked | Wojowu | CC BY-SA 4.0 |