Timeline for Are submersions of differentiable manifolds flat morphisms?
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28 events
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| Apr 20, 2020 at 14:16 | comment | added | Michael Bächtold | @mfox I don't think I used the flatness definition of Gonzales, Salas in my question. Does their definition of tensor product reproduce the classical pullback construction in the case where $E$ is the module of smooth sections of a vector bundle over a manifold $M$, and $\phi: N \to M$ is smooth map of manifolds? I.e. is $E\otimes_{\phi} C^\infty(N)$ isomorphic to the module of sections of the pulled back vector bundle with their definition of tensor product? | |
| Apr 20, 2020 at 13:54 | comment | added | mfox | An $R$-module M is flat if the functor $M\otimes- : R-Mod\rightarrow R-Mod$ preserves exact sequences; this is equivalent to the equational condition that you gave, but not to the definition of a flat module in the book by Gonzalez and Salas, simply because in their definition a module over a differentiable algebra is not just a module on its underlying ring, and the tensor product of such modules does not coincide with the usual one. This is what I wanted to say. | |
| Apr 19, 2020 at 10:06 | comment | added | Michael Bächtold | @mfox in the context I was interested in, it was not the tensor product of rings of smooth functions that I care about, but the tensor product of modules over the ring of smooth functions (sections of some vector bundles mainly). In that context the usual tensor product is appropriate and useful. In particular: if you have a sequence of vector bundles $E_i$ over a manifold $M$ and a smooth map from another manifold $N$ to $M$, then you get the pulled back sequence by tensoring the modules of sections with $C^\infty N$. | |
| Apr 18, 2020 at 18:38 | comment | added | mfox | It seems to me that the definition of flatness in Gonzales, Salas, C∞-differentiable spaces is not the same that you mentioned above, because the tensor product of Frechet modules over a Frechet algebra does not coincide with the tensor product of modules over a ring. Also, the coproduct in the category of C^{\infty} rings does not coincide with the coproduct in the category of rings. I believe that the two previous remarks should suggest that the notion of flatness that you are considering is not appropriate in this context. | |
| Jul 28, 2019 at 21:36 | answer | added | Michael | timeline score: 4 | |
| Jun 28, 2010 at 16:13 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Feb 19, 2010 at 0:40 | answer | added | George Lowther | timeline score: 12 | |
| Feb 18, 2010 at 21:17 | comment | added | Pete L. Clark | @DY: Fair enough. In my opinion, the language of sheaves is an appropriate one for all kinds of geometry. The main advantage is that it provides a unified perspective, so that indeed people can talk about smooth manifolds, complex analytic spaces, p-adic analytic varieties etc. in terms that will be familiar to lots of different types of geometers. I do think that it's mostly a linguistic advantage, though.... | |
| Feb 18, 2010 at 13:01 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Feb 18, 2010 at 3:44 | comment | added | Deane Yang | @Pete: Having had to live with algebraic geometers as a graduate student, I can definitely understand how the question could arise naturally from that perspective. I was curious, however, about whether anyone sees the possibility of using this approach to address the more commonly posed questions in differential geometry. There was a time in the 60's and 70's when sheaves and all that were all the rage, so even differential geometers tried to view their subject from that viewpoint You can see this in Frank Warner's book. But nothing came of it back then, and everybody abandoned the approach. | |
| Feb 17, 2010 at 23:17 | comment | added | Pete L. Clark | @DY: For someone who is coming at geometry from an algebraic perspective, this is a fairly natural question to ask (although I have not heard it before). The general line of inquiry goes back at least to Richard Swan's classic paper on the correspondence between vector bundles and projective modules over the ring of C^{oo} functions. | |
| Feb 17, 2010 at 19:35 | comment | added | Deane Yang | I have no idea about flatness, but does this approach to topology correspond to the "commutative" case of Connes' noncommutative differential geometry? | |
| Feb 17, 2010 at 18:23 | comment | added | Michael Bächtold | I've added a small section trying to motivate the question | |
| Feb 17, 2010 at 18:22 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Feb 17, 2010 at 1:55 | comment | added | Deane Yang | Could someone explain why this is an interesting or useful thing to know? | |
| Feb 17, 2010 at 0:24 | answer | added | George Lowther | timeline score: 12 | |
| Nov 23, 2009 at 21:50 | answer | added | Jorge Vitório Pereira | timeline score: 8 | |
| Nov 6, 2009 at 21:49 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Nov 3, 2009 at 22:01 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Oct 25, 2009 at 1:19 | answer | added | Greg Stevenson | timeline score: 3 | |
| Oct 24, 2009 at 21:16 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Oct 20, 2009 at 14:57 | history | edited | Michael Bächtold |
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| Oct 19, 2009 at 21:05 | comment | added | Michael Bächtold | Oh i guess html doesn't work in comments. What i meant was C^\infty(N) etc. | |
| Oct 19, 2009 at 21:03 | comment | added | Michael Bächtold | The flatness condition I gave is equivalent to saying that tensoring exact sequences of C<sup>&\infin;</sup>(N) modules with C<sup>&\infin;</sup>(M) gives exact sequences (a proof of this equivalence may be found in Eisenbud). I guess that is what you meant with your second question. I think this condition implies the local flatness you mention. I don't know if it is equivalent to it, but it might be useful having such an equivalence. | |
| Oct 19, 2009 at 16:29 | comment | added | David Zureick-Brown | Is your definition of flatness the same as saying that O_{M,x} is a flat O_{N,f(x)} module for every x in M? Or that p^* is exact on O_N modules? | |
| Oct 19, 2009 at 15:47 | history | edited | David Zureick-Brown | CC BY-SA 2.5 |
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| Oct 19, 2009 at 14:32 | history | edited | Michael Bächtold | CC BY-SA 2.5 |
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| Oct 19, 2009 at 14:16 | history | asked | Michael Bächtold | CC BY-SA 2.5 |