Timeline for What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 28, 2018 at 22:01 | vote | accept | dorebell | ||
| Sep 28, 2018 at 21:58 | answer | added | David Benjamin Lim | timeline score: 3 | |
| Sep 28, 2018 at 2:04 | comment | added | dorebell | Ah great! This is what I was hoping for - does anyone know what conditions on $\mathscr X$ make $\mathscr X_x = B \mathrm{Stab}_x$? Also, does proper base change work when $\ell \mid |G|$ (but isn't equal to the field characteristic)? I'd love to read a proof of these sorts of basic etale cohomology theorems which work for stacks. | |
| Sep 28, 2018 at 0:45 | comment | added | Will Sawin | Even in the case where $Y$ is a point, so $\mathcal X$ is $BG$, you will be in trouble if $\ell$ divides $|G|$, as the cohomology of $BG$ is group cohomology which won’t match. But this should be your only obstruction by proper base change under whatever hypotheses are needed to ensure $\mathcal X_x$ is a single point. | |
| Sep 27, 2018 at 22:34 | comment | added | skd | I don't have an answer, but just a comment regarding the analogous q'n for quasicoherent sheaves: if $\mathscr{X}$ is a tame DM-stack, then (by definition) the derived functors $\mathrm{R}^i \pi_\ast \mathscr{F}$ vanish for every quasicoherent sheaf $\mathscr{F}$ on $\mathscr{X}$ and every $i>0$, and any DM-stack over a field of characteristic $0$ is tame. | |
| Sep 27, 2018 at 20:16 | history | asked | dorebell | CC BY-SA 4.0 |