Timeline for How to classify the algebras C^∞(M)?
Current License: CC BY-SA 2.5
27 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 21, 2010 at 17:02 | vote | accept | Martin Brandenburg | ||
| Apr 15, 2010 at 18:15 | comment | added | Tim Perutz | "At first glance, it appears too optimistic to find an algebraic characterization. But many famous problems started like that and involved unexpected methods." I don't know how to convince you that this is the problem Connes has solved! He has characterised function algebras of compact smooth manifolds by saying that these algebras have a canonical projective module whose properties one can recognise. The cohomology of a compact manifold satisfies Poincare duality, and $C^\infty(M)$ knows this via the structure of its Hochschild/cyclic homology - hence Connes' invocation of Hochschild chains. | |
| Apr 14, 2010 at 23:39 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 14, 2010 at 15:04 | answer | added | Michael Bächtold | timeline score: 3 | |
| Apr 13, 2010 at 21:48 | comment | added | Martin Brandenburg | concerning the power series: I don't think that every formal power series arises as a function on $M$ ;). | |
| Apr 13, 2010 at 21:04 | comment | added | Martin Brandenburg | ok how do you prove that $C^{\infty}$ is not finitely generated (if $M$ is not discrete)? | |
| Apr 13, 2010 at 19:23 | comment | added | Michael Bächtold | @Martin: a polynomial algebra is finitely generated while smooth function algebras are not. I'm not sure of what you are aiming at with your comment about convergent power series. | |
| Apr 13, 2010 at 17:38 | answer | added | Tim Perutz | timeline score: 11 | |
| Apr 13, 2010 at 17:23 | comment | added | Martin Brandenburg | @unknown: a basic open set is $\{p : f(p) \neq 0\}$ the sections on it $A_f$ . @michael: right, we need some completeness of $A$. but I think that we also have to talk about concergent power series on open subsets, somehow ... btw. how do you prove that $R[x]$ is not an algebra of the form? | |
| Apr 13, 2010 at 16:41 | comment | added | Michael Bächtold | Correct me if I'm wrong: if you take a polynomial algebra over the reals then it satisfies all of your conditions but is not the algebra of functions of a smooth manifold. One additional condition might be that any formal power series (infinite jet) is realized as the taylor series of an element of your algebra. This is called Borel lemma for smooth functions. | |
| Apr 13, 2010 at 15:53 | comment | added | Qfwfq | (I assume $Hom(A,\mathbb{R})$ is the set of (unital) $R$-algebra homomorphisms from $A$ to $\mathbb{R}$) | |
| Apr 13, 2010 at 15:49 | comment | added | Qfwfq | Question: what is the "obvious structure sheaf" on $Hom(A,\mathbb{R})$ ? | |
| Apr 13, 2010 at 15:27 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Apr 13, 2010 at 11:03 | answer | added | Miguel | timeline score: -1 | |
| Apr 13, 2010 at 2:53 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 13, 2010 at 2:46 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 13, 2010 at 1:45 | answer | added | Dmitri Pavlov | timeline score: 10 | |
| Apr 13, 2010 at 1:44 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 13, 2010 at 1:02 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 13, 2010 at 0:56 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Apr 13, 2010 at 0:52 | comment | added | Martin Brandenburg | @Qiaochu Yuan: Yeah but I want to avoid it. I'm not sure if it works. @Jonas: Thank you, this is nice. But I think it's not that handy. In concrete examples you have already found a manifold representing the algebra before using the characterization, right? Well, perhaps its the best you can get ... | |
| Apr 13, 2010 at 0:47 | comment | added | Martin Brandenburg | @Jason: ok you already posted the same question (but already assuming $M$ to be compact) but the answer basically consists of abstraction typical for MO ... I want something concrete, you know. | |
| Apr 13, 2010 at 0:44 | comment | added | Jason DeVito - on hiatus | I asked something similar earlier - you may find the references there helpful. mathoverflow.net/questions/5344/… | |
| Apr 13, 2010 at 0:39 | comment | added | Jonas Meyer | Does this help? arxiv.org/abs/math.GT/9404228 | |
| Apr 13, 2010 at 0:38 | comment | added | Qiaochu Yuan | Shouldn't the right condition be something like "is locally isomorphic to C^{\infty}(U) for some open subset of R^n"? I don't know if one can avoid mentioning R^n somewhere in the answer... | |
| Apr 13, 2010 at 0:27 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |