Timeline for Why should I prefer bundles to (surjective) submersions?
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| Oct 14, 2022 at 21:22 | comment | added | Z. M | @SándorKovács More precisely, are you saying that Leray spectral sequences, for (quasi)coherent cohomology, behaves nice for flat maps? I think that, if you want to understand étale cohomology, you have to play with ULA maps instead of flat maps? | |
| Jan 24, 2011 at 5:16 | comment | added | Sándor Kovács | Ben, sorry, I did indeed have proper flat in mind with respect to that comment about Leray. And I am happy to call those fibrations. However, the original question was about bundles. But the main point of my comment was that I think that we still do see submersions on a regular basis, just don't call them that. | |
| Jan 24, 2011 at 4:46 | comment | added | Ben Webster♦ | I'm not sure what you mean by "the Leray spectral sequence works quite nicely for flat morphisms." If you take an arbitrary flat morphism (say the inclusion of a curve minus a point into a curve), a naive interpretation of Leray-Serre gives nonsense; of course this can be fixed, as Ryan points out, but at a significant cost in terms of complication. Of course, things work beautifully for proper smooth maps, but I would call those fibrations; they are in the analytic topology over $\mathbb C$, and behave like them over other fields. | |
| Jan 24, 2011 at 1:21 | comment | added | Sándor Kovács | Ben, I disagree with your statement that "algebraic geometers [...] will extremely rarely encounter submersions". We just call them "smooth morphisms". Also, the Leray spectral sequence behaves quite nicely already for flat morphisms, you don't even need smoothness. | |
| Aug 4, 2010 at 12:57 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Mar 11, 2010 at 21:10 | comment | added | Ben Webster♦ | It's a bit unfair to say "nothing useful," but at the same time, I'm not very good at taking cohomology of random constructible sheaves on a space, as opposed to the local systems that show up in Serre for a bundle. | |
| Mar 11, 2010 at 21:04 | comment | added | Ryan Budney | There is the Leray spectral sequence of a map. It's just much better behaved for a fibration. | |
| Mar 11, 2010 at 18:03 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |