Timeline for Why are flat morphisms "flat?"
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Aug 2, 2019 at 16:33 | comment | added | R. van Dobben de Bruyn | Rather than 'varying continuously', I would probably say 'nothing sticking out'. In the projective case, this is what the Hilbert polynomial criterion is getting at: all numerical measures of 'size' are the same across all fibres (dimension, degree, ...). (See also this answer I wrote on MSE for a similar question.) | |
| Nov 27, 2009 at 13:37 | comment | added | Kevin H. Lin | Anyway my point is just that one should be careful about specifying exactly what is meant by "continuous" in this situation, as it can easily be misleading. | |
| Nov 25, 2009 at 23:57 | comment | added | Kevin H. Lin | Dan: I don't like the description of flat families as families that "vary continuously". As you note, an important feature of flat families is that they allow degenerations. For me at least "continuously varying family" should mean that the topology doesn't change, but for instance $xy=0$ and $xy=1$ are topologically different. For intuition purposes, I prefer to think of smooth morphisms as being "continuous families", and flat morphisms as being "not-too-badly-discontinuous families". | |
| Nov 25, 2009 at 15:43 | comment | added | Daniel Erman | @Jose: To address your point, I added a sentence at the end of the first paragraph. | |
| Nov 25, 2009 at 15:42 | history | edited | Daniel Erman | CC BY-SA 2.5 |
Responded to a comment.
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| Nov 25, 2009 at 14:19 | comment | added | Jose Capco | but this does not answer the OP: what does "flat" the way we know it in everyday life have to do with "flat" the way we know it for algebras and schemes. | |
| Nov 25, 2009 at 14:02 | history | answered | Daniel Erman | CC BY-SA 2.5 |