Timeline for Why are flat morphisms "flat?"
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 26, 2009 at 8:55 | comment | added | Joel Fine | Thanks for your comments Kevin. I certainly didn't mean to suggest that flat families would be locally trivial. (Differential geometers like degenerations too!) I was more wondering if there was any sort of analogy between the two situations. E.g., for what families can one find a flat bundle on the base (more correctly one should probably look for a complex of sheaves, what ever flat means there) in analogy with Gauss-Manin? | |
| Nov 26, 2009 at 0:38 | comment | added | Kevin H. Lin | On the other hand, if we have an algebraic morphism which is smooth and proper in the sense of algebraic geometry, and if we consider that morphism as an analytic morphism, then it will be a proper submersion in the sense of differential geometry. Then Ehresmann's theorem says that proper submersion = locally trivial fiber bundle, and thus it follows that the topology of the fibers is constant (for the pedants: assuming the base is connected), and then we get the Gauss-Manin connection and all that other good stuff. But I don't know of any analogous thing for flat morphisms. | |
| Nov 26, 2009 at 0:08 | comment | added | Kevin H. Lin | This is not a complete response to your comments, but -- one of the main points of flatness is that it allows degenerations, and in particular it allows the topology of the fibers to change. For example you can have flat families with some fibers being smooth elliptic curves (topologically a torus) and other fibers being singular elliptic curves (topologically a "pinched" torus), and the (singular) cohomology will not be constant. | |
| Nov 25, 2009 at 13:15 | history | answered | Joel Fine | CC BY-SA 2.5 |