Timeline for Question regarding étale sheaf under finite étale surjective morphism
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 3, 2022 at 16:52 | comment | added | A.B. | @Hajime_Saito One possible reference is 5.8.1 of Etale cohomology theory by Lei Fu. See also 3.2.12 in loc. cit. | |
| Apr 3, 2022 at 15:33 | comment | added | Hajime_Saito | @PiotrAchinger Please suggest a reference for this correspondence... it would really help me. | |
| Apr 3, 2022 at 15:32 | comment | added | Hajime_Saito | @abx thanks for your comment. Please suggest a reference to read this correspondence that you and Piotr Achinger are pointing to... I have only found similar statements in the context of local systems or when the stalks are finite. I would really like to know in what generality it holds. | |
| Apr 3, 2022 at 9:30 | history | edited | YCor | CC BY-SA 4.0 |
fixed English
|
| Apr 3, 2022 at 6:39 | comment | added | Piotr Achinger | Such sheaves correspond to homomorphisms $G\to \mathrm{GL}_r(\mathbb{Z})$ and so are not necessarily constant. | |
| Apr 3, 2022 at 6:37 | comment | added | abx | No, $\mathscr{F}$ can be non-constant. In your case, there is an action of $G$ on $\mathbb{Z}^r$, and $\mathscr{F}$ is constant if and only if this action is trivial. For instance if $G=\mathbb{Z}/2$ acting on $\mathbb{Z}$ by changing sign, $\mathscr{F}$ is a rank 1 non-constant sheaf. | |
| Apr 3, 2022 at 6:07 | history | asked | Hajime_Saito | CC BY-SA 4.0 |