Timeline for Tannakian-type reconstruction of étale fundamental group
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2022 at 19:49 | comment | added | Will Sawin | @Niels Sure, or the enlarged fundamental group defined in the Bhatt-Scholze pro-étale topology paper. | |
| Jul 23, 2022 at 15:18 | comment | added | Niels | in the second paragraph, you may want to consider the finite-étale topology to avoid discussing normality. Alternatively, you may keep the étale topology and introduce SGA 3 enlarged fundamental group | |
| Jul 22, 2022 at 18:55 | comment | added | Will Sawin | @Vite I don't know a reference (maybe the Nori paper naf mentions discusses it), but it's not hard to prove. I'm not saying the Tannakian groups are necessarily different, but more that it would be very difficult to prove they are isomorphic (while it's easy to prove the component groups are isomorphic.) Conjecturally, a quotient of the $\ell$-adic Tannakian group should be the base change to $\mathbb Q_\ell$ of the motivic Galois group of the category of locally constant motives, whatever that means. | |
| Jul 22, 2022 at 18:36 | comment | added | Vite | @WillSawin, Are the different $l$-adic Tannakian groups related to some "motivic" group? | |
| Jul 22, 2022 at 18:28 | comment | added | Vite | @WillSawin, What is a good reference for the reconstruction from lisse $l$-adic sheaves? In the last paragraph, do you mean that the Tannakian pro-algebraic groups for different $l$ are different, but their component groups are always isomorphic to $\pi_1$? Thanks! | |
| Jul 22, 2022 at 18:23 | vote | accept | Vite | ||
| Jul 22, 2022 at 16:51 | history | edited | Will Sawin | CC BY-SA 4.0 |
added 13 characters in body
|
| Jul 22, 2022 at 16:12 | comment | added | Niels | in your second paragraph when you mention locally constant sheaves you should specify the topology else it doesn't really make sense | |
| Jul 22, 2022 at 10:46 | comment | added | Will Sawin | @Z.M One doesn't have to go to topos theory or pyknotic stuff - it's just the category of finite étale covers. Yes, as long as one considers finite étale covers to be purely topological (they're not always determined by the Zariski topology!), the étale fundamental group is a purely topological notion. When I say "algebraic", I literally mean as an algebraic group. | |
| Jul 22, 2022 at 3:41 | comment | added | Z. M | I am not sure how this algebraic structure relates to the "algebraic structure" on X. It seems to be constructed out of the (pro)étale topos of X, or the Galois category à la Barwick–Haine, which seems to contain only "topological" data. | |
| Jul 21, 2022 at 18:54 | history | answered | Will Sawin | CC BY-SA 4.0 |