Timeline for Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2022 at 16:23 | comment | added | Jesse Silliman | @PrimeRibeyeDeal Yes, I meant $\pi^{-1} \mathcal{O}_S$, and you are right that some sort of hypothesis, such as normal, is necessary. If $S = \mathrm{Spec}(\mathbb{C})$, this type of object is a sheaf whose analytification is a $\mathbb{C}$-local system on $X^{\mathrm{an}}$ with finite monodromy. | |
| Mar 3, 2022 at 15:55 | comment | added | Will Sawin | I'm not sure what is needed exactly but normal sounds right. | |
| Mar 3, 2022 at 15:54 | comment | added | PrimeRibeyeDeal | @JesseSilliman I'm unfamiliar with that type of object. Do you have any suggestions for how to prove something like that? And should that be finite rank $\pi^{-1}\mathcal{O}_S$-module? | |
| Mar 3, 2022 at 15:51 | comment | added | PrimeRibeyeDeal | @WillSawin Thank you, good idea. Pardon my ignorance, but what do I need to make that work? $X$ normal, to use Weil divisors and valuation rings? | |
| Mar 3, 2022 at 5:31 | comment | added | Jesse Silliman | @WillSawin I think I should have said that I'm guessing that $\mathcal{E}^{\nabla}$ is a finite rank $\pi^*\mathcal{O}_S$-module on $X$ which becomes a "constant", i.e. pulled back from on $S$, after a finite etale covering $X' \rightarrow X$. This is not a Nori finite vector bundle anymore, though. | |
| Mar 3, 2022 at 3:27 | comment | added | Will Sawin | Probably you can check that this sheaf satisfies a relative form of flasqueness, where for $U \subset V$ open sets, if the image of $U$ and $V$ in $S$ is the same, then the restriction map is surjective. The point is to consider a section on $U$, which has a pole on some component of $V \setminus U$, and check that the derivative must have a pole of higher order to get a contradiction. Then that claim should suffice for vanishing of the higher direct image. | |
| Mar 3, 2022 at 3:13 | comment | added | Will Sawin | @JesseSilliman I don't see why it should be trivialized. Can't we take the pullback of any vector bundle and use the standard connection on the fibers (since it's the trivial bundle on the fibers)? But the distinction may not be so relevant for this problem. | |
| Mar 3, 2022 at 3:06 | history | edited | PrimeRibeyeDeal | CC BY-SA 4.0 |
added condition
|
| Mar 3, 2022 at 3:04 | comment | added | PrimeRibeyeDeal | @JesseSilliman Thank you for the suggestion. Can you recommend a reference for that topic? Also, the characteristic is an important detail I neglected to mention. | |
| Mar 3, 2022 at 1:31 | comment | added | Jesse Silliman | I would guess that, at least in characteristic 0, $\mathcal{E}^{\nabla}$ is a vector bundle on $S$ which is trivialized by a finite etale covering $S' \rightarrow S$ (i.e. a finite vector bundle in the sense of Nori). | |
| Mar 2, 2022 at 22:50 | history | asked | PrimeRibeyeDeal | CC BY-SA 4.0 |