Timeline for Cohomological dimension for stacks
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 17, 2021 at 10:19 | comment | added | Pulcinella | @JasonStarr I've just realised I don't actually understand the argument that $R^if_*\mathbf{Q}_\ell=0$ for $i>0$ and $f$ is the coarse moduli space of a DM stack. Is there a proof of this written anywhere? | |
| Mar 23, 2021 at 16:23 | comment | added | Jason Starr | I explained this to a graduate student a couple of weeks ago. There is a quasi-finite, proper $1$-morphism from the stack to a "coarse moduli space". Etale locally, this morphism is the coarse moduli space of a finite quotient stack (you can find a proof in Abramovich-Vistoli). Thus, by the argument in the previous comment, the higher direct images vanish for "rational" $\mathbb{Q}_\ell$-coefficients. Combined with the Leray spectral sequence, cohomological dimension of the stack equals cohomological dimension of the coarse moduli space. | |
| Mar 23, 2021 at 10:56 | comment | added | user42024 | but I can't recall the argument | |
| Mar 23, 2021 at 10:56 | comment | added | user42024 | And it is also necessary to make assumptions on the cohomological dimension of the base field! Let's maybe assume the base field is algebraically closed. Then I think the statement is true for all DM-stacks $X$ of finite type. This is definitely true for quotient stacks by etale group schemes: you get a spectral sequence converging from the cohomology of a (finite) group with coefficients in the cohomology of a finite type scheme.. I guess it should also be true that if $U\rightarrow X$ is an etale cover, then rat. cohomological dimension of $X$ is bounded by the cohomological dimension of $U$ | |
| Mar 23, 2021 at 10:50 | comment | added | user42024 | @Faris Yeah, I guess $X$ should should be of finite type | |
| Mar 23, 2021 at 10:47 | comment | added | Faris | What if $X$ is the disjoint union of complex projective spaces of each dimension? | |
| Mar 23, 2021 at 10:09 | history | asked | Pulcinella | CC BY-SA 4.0 |