Timeline for Fundamental group of formal punctured disc and punctured affine line
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2019 at 0:33 | comment | added | Ben Wieland | If you know that $\mathbb A^1$ and the formal disk are simply connected (requiring characteristic 0), then van Kampen should tell you that the Galois group of the formal disk surjects to the Galois group of $\mathbb A^1$. | |
| Jun 25, 2019 at 17:43 | comment | added | Jędrzej Garnek | @StevenLandsburg I will definitely ask the person, as soon as I will get touch with him. But I'm afraid that he meant a hand-waving argument rather then something formal. | |
| Jun 25, 2019 at 14:58 | comment | added | Steven Landsburg | Have you tried asking the person who handed you the problem? | |
| Jun 25, 2019 at 13:17 | comment | added | user108998 | Minor comment, we have an equiv of etale ho-types from an n-disc to affine n-space. We'd like to "remove a point". The result is that in this case this is valid, it certainly isn't in general, it'd be nice to know when this sort of reasoning can be applied. | |
| Jun 25, 2019 at 11:41 | history | edited | Mike Shulman |
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| Jun 25, 2019 at 7:23 | comment | added | Jędrzej Garnek | @TheoJohnson-Freyd: I thought of using the hint the way you suggested, but it doesn't seem obvious that the covers of the punctered disc extend uniquely. Besides, exactly as you mentioned, I would be more satisfied with the "étale homotopy type" statement. | |
| Jun 25, 2019 at 0:42 | comment | added | Theo Johnson-Freyd | I'm not an algebraic geometer. I think the hint was intended as: an etale cover of $\mathbb{A}^1 \setminus \{0\}$ restricts to an etale cover of $\mathrm{Spec}(\mathbb{C}((t))) = \mathbb{D}^1 \setminus \{0\}$; now classify those, and show that each one extends uniquely to $\mathbb{A}^1 \setminus \{0\}$. But this is a disappointing interpretation: as you point out, it is more satisfying to have a general criterion that says that the inclusion $\mathbb{D}^k \setminus\{0\} \subset \mathbb{A}^k \setminus\{0\}$ is an etale homotopy equivalence. | |
| Jun 25, 2019 at 0:39 | comment | added | Theo Johnson-Freyd | +1, especially for the subquestion "Does it generalize to higher dimensions?", which I interpret as asking about the etale homotopy type of $\mathbb{A}^n \setminus \{0\}$ versus its formal completion near zero. | |
| Jun 24, 2019 at 20:43 | history | asked | Jędrzej Garnek | CC BY-SA 4.0 |