Timeline for Is there a "Hilbert syzygy theorem" for smooth manifolds? Or: does every finitely generated $C^\infty$ module have a finite-length resolution in vector bundles?
Current License: CC BY-SA 2.5
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| Oct 7, 2010 at 10:24 | comment | added | Tom Goodwillie | Yes, that seems like a good way to summarize the situation. | |
| Oct 7, 2010 at 5:01 | comment | added | Dave Anderson | Doesn't the example given in the edit essentially show that $C^\infty(X)$ is not coherent as a sheaf of rings? I've heard it said that Serre gave this counterexample when developing the theory of coherent sheaves, but can't find it in FAC. Anyway, this kind of property -- the structure sheaf not being coherent -- seems to rule out translating a lot of the sheafy power of tools from analytic or algebraic geometry over to differential topology. | |
| Oct 7, 2010 at 2:54 | history | edited | Tom Goodwillie | CC BY-SA 2.5 |
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| Oct 7, 2010 at 2:17 | comment | added | Tom Goodwillie | The answer is still "no". I'll explain in an edit. | |
| Oct 7, 2010 at 1:18 | comment | added | Theo Johnson-Freyd | Awesome. Do you know if the answer is "yes" if I insist that $M$ be a submodule of (the sheaf of sections of) some vector bundle? Or I can ask that as another MO question... | |
| Oct 7, 2010 at 1:14 | vote | accept | Theo Johnson-Freyd | ||
| Oct 7, 2010 at 0:40 | history | edited | Tom Goodwillie | CC BY-SA 2.5 |
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| Oct 7, 2010 at 0:20 | history | answered | Tom Goodwillie | CC BY-SA 2.5 |