Timeline for Are submersions of differentiable manifolds flat morphisms?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2010 at 2:42 | comment | added | Greg Stevenson | @fpqc: Yeh, I had that in mind when I wrote this (I think... it was a while ago) but I was entertaining the vain hope that one might somehow manage to push it through anyway. | |
| Feb 17, 2010 at 0:48 | comment | added | Harry Gindi | Oh, haha, that's what the comment above mine says! | |
| Feb 17, 2010 at 0:47 | comment | added | Harry Gindi | @Greg: In general, the stalks of the sheaf of a manifold are not noetherian, so Idon't believe it will work. | |
| Oct 27, 2009 at 14:04 | comment | added | Michael Bächtold | So everything would go trough, except for the "local criterion of flatness" (Eisenbud Ch 6.4) which assumes notherianity. I've had a look at two proofs of it and they both rely on Krulls theorem applied to the maximal ideal in the germ of functions. That doesn't work here because of "flat smooth functions". For the moment I don't see an easy way to adapt the proof. | |
| Oct 25, 2009 at 20:21 | comment | added | Greg Stevenson | A reference is Hartshorne Ch III Prop 10.4 (the proof of (iii) implies (i)). The basic idea is that identifying the stalks of the cotangent sheaf with m/m^2 one can lift a basis to a regular sequence at x and f(x) and injectivity says the sequence at f(x) maps injectively into the one at x. One can take the quotient at f(x) by the max ideal and by the corresponding image at x which is flat since the quotient at f(x) is the residue field. Then one uses the local criterion to finish by induction. | |
| Oct 25, 2009 at 17:00 | comment | added | Michael Bächtold | Thanks i'll have a look at that. Could you give me a reference for that argument of translating injectivity into regular sequences? | |
| Oct 25, 2009 at 1:32 | history | edited | Greg Stevenson | CC BY-SA 2.5 |
objects -> fields
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| Oct 25, 2009 at 1:19 | history | answered | Greg Stevenson | CC BY-SA 2.5 |