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.
Best Answer
(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:
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?