Timeline for action of étale fundamental group on the cover
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 12, 2019 at 2:40 | comment | added | Will Sawin | The key point is that given a group $G$, acting on a set $S$ simply transitively, the automorphisms of $S$ that commute with the action of $G$ are themselves isomorphic to $G$. Here $G$ is the group of deck transformations, which acts simply transitively on $Y_{\overline{x}}$, and which the automorphisms of the fiber functor must commute with, showing that the automorphisms of the fiber functor map to the group of deck transformations. That's all that's going on. | |
| Dec 11, 2019 at 20:48 | comment | added | Denis Nardin | Unless I'm mistaken, the definition I gave is the one in construction 3.15 of Lenstra's Galois theory for schemes | |
| Dec 11, 2019 at 20:35 | comment | added | Anh Dũng Lê | @Santana Afton I find it difficult to analogize, since in topology I can think with actual loops and universal cover but I cannot do it here in algebraic geometry. | |
| Dec 11, 2019 at 20:34 | comment | added | Anh Dũng Lê | @Denis Nardin I am studying with Stacks project and Tamas Szamuely's Galois groups and fundamental groups and both of them only have the definition with fiber functor. | |
| Dec 11, 2019 at 19:59 | comment | added | Santana Afton | You should be heavily analogizing with algebraic topology at every step. The definitions given by Denis Nardin are equivalent because the fiber, seen as a $\pi_1$-set, has the same automorphism group as the cover does — namely, the deck group. | |
| Dec 11, 2019 at 17:48 | comment | added | Denis Nardin | @AnhDungLe Yes they are equivalent. I recommend studying Grothendieck Galois theory before asking questions about the topic on MO. | |
| Dec 11, 2019 at 17:46 | comment | added | Anh Dũng Lê | I thought the fundamental group is the automorphism group $Aut(F_{\bar{x}})$ where $F_{\bar{x}}$ is the functor from finite etale morphisms over $X$ to sets by taking the geometric fiber of $\bar{X}$. Are these definitions equivalent? | |
| Dec 11, 2019 at 17:28 | comment | added | Denis Nardin | The étale fundamental group comes by definition with a canonical projection $\pi_1^{ét}(X,\bar x)\to \mathrm{Aut}(Y/X)$. In fact it is defined as the limit of those automorphism groups as $Y$ runs through the Galois covers... | |
| Dec 11, 2019 at 16:35 | history | edited | Anh Dũng Lê | CC BY-SA 4.0 |
added 99 characters in body
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| Dec 11, 2019 at 16:29 | history | undeleted | Anh Dũng Lê | ||
| Dec 11, 2019 at 16:29 | history | deleted | Anh Dũng Lê | via Vote | |
| Dec 11, 2019 at 16:25 | history | asked | Anh Dũng Lê | CC BY-SA 4.0 |