Timeline for Residue field of point on an algebraic stack
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 4, 2021 at 19:23 | vote | accept | Daniel Loughran | ||
| Aug 3, 2021 at 20:32 | comment | added | Daniel Loughran | Nice. For a moduli stack, I think the residue field is the field of moduli, then an example would be a variety whose field of moduli is not a field of definition. | |
| Aug 3, 2021 at 19:32 | comment | added | Will Sawin | @DanielLoughran Correct. I think an example where this will not exist, because there is no minimal field as in 1, is a gerbe over $\mathbb Q$ or another nice field associated to a nontrivial class in $H^2(\mathbb Q, \mu_2)$. There are many extensions of degree $2$ over which this gerbe trivializes and hence the stack has a point, but no extension of lesser degree, and thus no minimal one. | |
| Aug 3, 2021 at 18:33 | comment | added | Daniel Loughran | Very slick construction, thanks. Just so I completely understand: your definition of the residue field $\kappa(x)$ does not necessarily come equipped with a morphism whose image is $x$? | |
| Aug 3, 2021 at 15:44 | comment | added | Will Sawin | I guess this is equivalent to the global sections of the structure sheaf on the residual gerbe. | |
| Aug 3, 2021 at 15:41 | comment | added | Will Sawin | @Z.M This notion of the residue field exists for any presheaf of groupoids. | |
| Aug 3, 2021 at 15:09 | comment | added | Z. M | Let me clarify: where do you use the fact that $X$ is an algebraic stack? Or you show that the residue field exists for any presheaf of groupoids? | |
| Aug 3, 2021 at 14:13 | history | answered | Will Sawin | CC BY-SA 4.0 |