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I noticed the following strange (to me) fact. If $M$ is a real manifold (smooth or not) and $R = C(X, \mathbb{R})$ is the ring of real functions (smooth functions in the smooth case) then the affine scheme $X = \mathrm{Spec}(R)$ has a natural map $M \to X$ which is a homeomorphism on real points i.e. $M \to X(\mathbb{R})$ is a homeomorphism. Even better, in the smooth case, we can identify $M$ with the ringed space $(X(\mathbb{R}), \mathcal{O}_X)$. Even more better, the functor $M \mapsto \mathrm{Spec}(C(M, \mathbb{R}))$ from the category of smooth manifolds to the category of affine schemes is fully faithful.

Add to this the Serre-Swan theorem which states that there is an equivalence between the category of vector bundles on $M$ and the category of finite projective $R$-modules i.e. vector bundles on $X$.

These facts seem to imply that smooth manifolds may be thought of "as" affine schemes. This observation leads me to ask the following questions:

(1) Do you know of any fruitful consequences or applications of looking at manifolds in this light?

(2) Is there anywhere this identification fundamentally fails?

(3) Is there an algebraic classification for what these rings look like? In particular, if $A$ is an $\mathbb{R}$-algebra then when is $X(\mathbb{R})$ a topological manifold and when can $X(\mathbb{R})$ be given a smooth structure compatible with $\mathcal{O}_{X}$?

(4) What do the "extra" points of $X$ look like? Is there a use for these extra points in manifold theory, the way that generic points have become important in algebraic geometry?

For question (4), I believe that maximal ideals of $R$ should correspond to ultrafilters on $M$ identifying the closed points of $X$ with the Stone-Cech compactification of $M$. What about the other prime ideals?

Many thanks.

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    $\begingroup$ The main issue is that commutative algebra doesn't tell you very interesting things about the spectrum of the kinds of commutative algebras over $\mathbb{R}$ that arise as algebras of functions on manifolds. But, this is a nice question, which hopefully someone more qualified than me will answer. $\endgroup$ – Andy Sanders Apr 2 at 22:23
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    $\begingroup$ arxiv.org/abs/1104.4951 $\endgroup$ – Zach Teitler Apr 2 at 23:56
  • $\begingroup$ (1): Serre-Swan makes thinking about vector bundles in the context of K-theory much more natural, at least for me. (I can't summarise that relation more effectively than most references you can search for can.) $\endgroup$ – AlexArvanitakis Apr 3 at 0:25
  • $\begingroup$ The book Nestruev, Smooth Manifolds and Observables is precisely about this algebraic perspective on manifolds. There you also find a characterization of the commutative algebras that are isomorphic to rings of smooth functions on manifolds. $\endgroup$ – Michael Bächtold Apr 22 at 21:16
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(1) This is a highly productive way of looking at smooth manifolds. It is responsible for synthetic differential geometry and derived smooth manifolds. Both of these subjects heavily rely on this identification.

Synthetic differential geometry looks at spectra of finite-dimensional real algebras, and interprets these as geometric spaces of infinitesimal shape. This allows one to make infinitesimal arguments of the type used by Élie Cartan and Sophus Lie perfectly rigorous. (Text)books have been written about this approach:

  • Anders Kock: Synthetic differential geometry;
  • Anders Kock: Synthetic geometry of manifolds;
  • René Lavendhomme: Basic concepts of synthetic differential geometry;
  • Ieke Moerdijk, Gonzalo Reyes: Models for smooth infinitesimal analysis.

Derived smooth manifolds start by enlarging the category of real algebras of smooth functions to a bigger, cocomplete category (such as all real algebras, or, better, C^∞-rings, see below). One then takes simplicial objects in this category of algebras, i.e., simplicial real algebras. This category admits a good theory of homotopy colimits, which are given by derived tensor products and other constructions from homological algebra. The opposite category of simplicial real algebras should be thought of as a category of geometric spaces of some kind. This category has an excellent theory of homotopy limits, in particular, one compute correct (i.e., derived) intersections of nontransversal submanifolds in it, and get expected answers. For example, this can be used to define canonical representative of characteristic classes, one could take the Euler class to be the derived zero locus of the zero section, for example. Other (potential) applications include the Fukaya ∞-category, which in presence of nontransversal intersections can be turned into a category in a more straightforward way.

Dominic Joyce wrote a book about this:

Other sources:

(2) Yes, for example, Kähler differentials are not smooth differential 1-forms, even though derivations are smooth vector fields. This is (one of) the reasons for passing to C^∞-rings instead. In the category of C^∞-rings, C^∞-Kähler differentials are precisely smooth differential 1-forms.

(3) Not really, this question was discussed here before, here is one of the discussions: Algebraic description of compact smooth manifolds?

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  • $\begingroup$ Dmitri, while sometimes useful, and maybe even occasionally essential (especially to build moduli spaces), do you really mean highly productive? I mean, any student of algebraic geometry would be ignored today if they didn't think of schemes in this fashion, but your average student studying most aspects of smooth manifolds probably doesn't even know that the theory you describe exists, ditto for their advisors. I'm not trying to be negative, and I think this perspective is important. I hope I don't give the opposite impression. $\endgroup$ – Andy Sanders Apr 3 at 17:53
  • $\begingroup$ @AndySanders: "Highly productive" is not the same thing as "well-known". My meaning is quite literal, that this approach produced a lot of good research. I did not say that the majority of those who work with smooth manifolds are familiar with it. $\endgroup$ – Dmitri Pavlov Apr 3 at 18:08
  • $\begingroup$ I guess we have different definitions of highly, but it's not worth arguing semantics. Regardless, thank you for this detailed and helpful post. I agree with your statement that it has produced good research. $\endgroup$ – Andy Sanders Apr 3 at 18:22
  • $\begingroup$ Thank you @DmitriPavlov for the detailed answer. I clearly have a lot of reading here to do. I do have a question about your remark regarding the Kahler differentials. $\endgroup$ – Ben C Apr 4 at 18:44
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    $\begingroup$ @BenC: The question about Kähler differentials for smooth functions is considered in great detail here: mathoverflow.net/questions/6074/…. $\endgroup$ – Dmitri Pavlov Apr 4 at 22:43

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