Timeline for $\mathbb{P}^n$ is simply connected
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2011 at 14:35 | vote | accept | Martin Brandenburg | ||
| Apr 24, 2011 at 15:56 | history | edited | Emerton | CC BY-SA 3.0 |
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| Apr 24, 2011 at 15:49 | comment | added | Emerton | Dear Martin, I have added a sketch of one argument (probably unnecessarily complicated) as to why this map is algebraic (i.e. a map of varieties, or of schemes). Regards, Matthew | |
| Apr 24, 2011 at 8:58 | comment | added | Martin Brandenburg | @Torsten: Thanks you! This is Theorem 30.6.2 in the stacks project. / Now my remaining question is why Emerton's map is actually a morphism of schemes. | |
| Apr 23, 2011 at 9:35 | comment | added | Torsten Wedhorn | @Martin: Any section of a separated morphism is a closed immersion. Any section of an unramified morphism is an open immersion. In particular any section of a finite etale morphism is an open and closed immersion. | |
| Apr 21, 2011 at 16:35 | comment | added | Martin Brandenburg | NB: I am not claiming that anything is missing here. Rather I have to admit that I don't understand basic arguments in projective algebraic geometry ... | |
| Apr 21, 2011 at 16:34 | comment | added | Martin Brandenburg | "any section of a finite etale morphism over a connected base induces an isomorphism between the base and a connected component of the cover." Why? Also, why is your set-map (yes I knew that we map $x$ to $y$) a morphism? | |
| Apr 21, 2011 at 16:21 | history | edited | Emerton | CC BY-SA 3.0 |
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| Apr 21, 2011 at 16:20 | comment | added | Emerton | Also, the map extends to $\mathbb P^n$ just by sending $x$ to $y$. | |
| Apr 21, 2011 at 16:17 | comment | added | Emerton | There is no need to assume that $Y$ is connected (although of course it is harmless to do so): any section of a finite etale morphism over a connected base induces an isomorphism between the base and a connected component of the cover. Thus the statement that a connected scheme is simply connected is equivalent to the statement that any finite etale morphism admits a section. | |
| Apr 21, 2011 at 15:47 | comment | added | BS. | @Martin : as in Sandor's answer, $Y$ must be silently assumed connected, so that a section is enough. Also, I have edited my answer, which seems to be a more cumbersome version of the same idea, but seemingly not needing extension. Does it seem more airtight to you ? I surmised that such simple geometric reasoning cannot let you out of algebraic geometry over anything, although I never felt so much assured with coverings in positive characteritic. | |
| Apr 21, 2011 at 8:22 | comment | added | Martin Brandenburg | This seems to be the most elementary approach, but still there are some details I don't understand. Namely, $x' \mapsto y'$ is first defined only as a set-theoretical map $\mathbb{P}^n \backslash x \to X \backslash x$. Why is it a morphism? And why can we extend it on $\mathbb{P}^n$? Why does it suffice to find a section? | |
| Apr 21, 2011 at 7:18 | history | edited | Emerton | CC BY-SA 3.0 |
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| Apr 20, 2011 at 13:06 | history | answered | Emerton | CC BY-SA 3.0 |