Timeline for Étale cohomology and normalization?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 27, 2020 at 12:16 | comment | added | David Hansen | Why say "$\infty$-topos" when you can just say "site"? | |
| S Jul 26, 2020 at 9:20 | history | edited | Glorfindel | CC BY-SA 4.0 |
typos corrected
|
| S Jul 26, 2020 at 9:20 | history | suggested | RobPratt | CC BY-SA 4.0 |
Corrected spelling in title
|
| Jul 26, 2020 at 9:06 | review | Suggested edits | |||
| S Jul 26, 2020 at 9:20 | |||||
| Jul 26, 2020 at 1:46 | comment | added | Harry Gindi | If I remember correctly, there is actually an equivalence for étale cohomology between X and its absolute weak normalization. That's because the map $X^{awn}\to X$ induces an equivalence of étale ∞-toposes. Over a field of characteristic $0$, this is the seminormalization, and over a field of characteristic $p>0$, it is the perfection. I think in general this is the best you can do. | |
| Jul 26, 2020 at 1:16 | comment | added | curious math guy | So, for example, if the closed subset wher $f$ is disconnected is a single point for instance, that would imply that for $i\geq 2$, we would have $H^i_{ét}(X,\mathbb{Z}/n)=H^i_{ét}(X^{norm},\mathbb{Z}/n)$, right? | |
| Jul 26, 2020 at 1:09 | comment | added | Will Sawin | The best way would be to understand the cokernel of the map. $\mathbb Z/n \to f_* \mathbb Z/n$. This is supported in the closed subset of $X$ where the fiber of $f$ is disconnected. So a first step would be to identify that closed subset. | |
| Jul 26, 2020 at 0:52 | history | edited | curious math guy | CC BY-SA 4.0 |
Adapted question after comment pointed out a mistake
|
| Jul 26, 2020 at 0:49 | comment | added | curious math guy | Ah, good, that makes sense, thank you. Is there a way to relate the $\ell$-adic cohomology of the normalization of a curve with the $\ell$-adic cohomology of the curve itself? | |
| Jul 25, 2020 at 23:56 | comment | added | dhy | I'm afraid that this result is not true (it fails for e.g. a nodal curve. One way to see this is to compute the singular cohomology over $\mathbb{C}$.) The issue is that the math.SE link you give is only for the Zariski topology, not the etale topology (see the comment of Roland.) | |
| Jul 25, 2020 at 23:23 | history | asked | curious math guy | CC BY-SA 4.0 |