Timeline for Philosophy : seeking examples illuminating deeper geometric ideas behind base change of schemes
Current License: CC BY-SA 2.5
9 events
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| Apr 27, 2010 at 18:20 | history | edited | xuros | CC BY-SA 2.5 |
improved statement
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| Apr 27, 2010 at 18:16 | answer | added | Steven Gubkin | timeline score: 3 | |
| Apr 27, 2010 at 15:25 | answer | added | Georges Elencwajg | timeline score: 5 | |
| Apr 27, 2010 at 14:38 | comment | added | Steven Gubkin | I second the idea of "fiber products". There are really two sides to the story. You have mentioned the algebra - if I am changing my base field, then I am "extending scalars", but there is no geometry here they are both fibers over a point. So I would recommend working through what pullbacks look like geometrically e.g. in the category of topological spaces. Pullbacks of schemes are funny because they incorperate both algebra and geometry. I still have not completely wrapped my head around them. | |
| Apr 27, 2010 at 14:18 | history | edited | xuros | CC BY-SA 2.5 |
edited title
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| Apr 27, 2010 at 13:45 | comment | added | xuros | Bases are also sort of parametrization (via the fibres), so I imagine base change could also be seen as a change of parametrization. | |
| Apr 27, 2010 at 13:43 | comment | added | xuros | I just meant the notion of base of a scheme is basically a high school idea ; to do things like multiplication/derivatives of polynomials, the operations are taking place over the 'smallest' ring containing the coefficients. Similarly, 'drawing' a curve over different fields amounts to taking points of a scheme, which is the same as looking at the sections of the projection of the base change. So base change as in the first paragraph (hard work aside) comes from HS ideas, whereas the étale coverings thing is more subtle. I'm just pointing out the fundamental difference in the two remarks. | |
| Apr 27, 2010 at 13:31 | comment | added | David E Speyer | What high school did you go to?!?! | |
| Apr 27, 2010 at 13:27 | history | asked | xuros | CC BY-SA 2.5 |