Timeline for Maps that admit local sections through each 'point' in the domain
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| May 30, 2019 at 21:05 | comment | added | David Roberts♦ | @Arrow ok, seen it :-) | |
| May 19, 2019 at 22:11 | comment | added | David Roberts♦ | @Arrow - good question! Why not ask it as a stand-alone Q? | |
| Jan 16, 2019 at 23:13 | comment | added | David Roberts♦ | @Arrow aha, got me. There are in fact two different definitions of topological submersion in the literature: one is 'admits local sections through every point in the domain', the other is 'looks locally like a projection'. For finite dimensional manifolds these coincide, and the second one implies local [homeo,diffeo]morphism. | |
| Jan 16, 2019 at 21:48 | comment | added | Arrow | Dear @DavidRoberts, but then these are topological submersions with discrete fibers which are not local homeomorphisms, contrary to the assertion at the end of the second paragraph. What am I missing? | |
| Jan 16, 2019 at 21:39 | comment | added | David Roberts♦ | @Arrow they indeed look to me to be topological submersions! | |
| Jan 16, 2019 at 14:32 | comment | added | Arrow | Dear @DavidRoberts, the map $\mathbb R\overset{x\mapsto x^2}{\longrightarrow}[0,\infty)$ is continuous with discrete fibers yet not a local homeomorphism about $0\in \mathbb R$. Why isn't it a topological submersion? It seems the only problem is $0\in \mathbb R$, but I think we can take the positive branch of the square root as a section. Same for the projection from the vanishing locus $(x-y)(x+y)=0$ onto the $x$-axis. (Sorry if I'm missing something silly.) | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 20, 2010 at 6:18 | history | edited | Bjørn Kjos-Hanssen |
edited tags
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| Sep 30, 2010 at 15:54 | answer | added | Emerton | timeline score: 3 | |
| Sep 29, 2010 at 9:28 | comment | added | David Roberts♦ | I'm a complete novice when it comes to alg.geom., so this is just the sort of information I am after. A short (but unsatisfactory) answer would then be 'no, this doesn't generalise from manifolds to schemes', but I presume that there are nice situations when something like this idea has merit? | |
| Sep 29, 2010 at 8:57 | comment | added | BCnrd | Consider the affine line $X$ over $Y = {\rm{Spec}}(k)$ for an imperfect field $k$. What do you propose to do for closed points $x \in X$ such that $k(x)/k$ is not separable? Demanding sections through all points of the sources seems much too strong if one wishes to work with the etale topology. (The analogy with manifolds misses some aspects.) In practice it is very often used that smooth maps admit etale-local sections, and fppf maps admit quasi-finite flat sections, but neither can be expected to pass through an arbitrary point in a fiber in general. | |
| Sep 29, 2010 at 6:04 | history | asked | David Roberts♦ | CC BY-SA 2.5 |