This continues my question about [smooth Gelfand-duality](https://mathoverflow.net/questions/21090/smooth-gelfand-duality). In the book



> 
> Juan A. Navarro González & Juan B. Sancho de Salas, C∞-Differentiable Spaces, LNM 1824
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> 
> 


it is shown that $M \mapsto C^\infty(M)$ is a fully faithfull contravariant functor from the category of manifolds (smooth, separable and without boundary) to the category of $\mathbb{R}$-algebras. Isn't this nice? It would be even more nice if there is an algebraic description of the essential image of this functor, so that we have an antiequivalence of categories between manifolds and certain $\mathbb{R}$-algebras. Thus my question is:


* Which $\mathbb{R}$-algebras $A$ are isomorphic to $C^{\infty}(M)$ for some manifold $M$?


Of course, you could just formulate that $Spec\_r(A)=Hom(A,\mathbb{R})$ with the obvious structure sheaf is a manifold and that the canonical map $A \to C^{\infty}(M)$ is an isomorphism in terms of the ring structure of $A$. But this does not seem to be handy at all. I want some nontrivial purely algebraic formulation. If possible avoiding structure sheaves at all.


Here are some necessary conditions:


* If $f \neq g$ in $A$, then there is some $\mathbb{R}$-homomorphism $\phi : A \to \mathbb{R}$ such that $\phi(f) \neq \phi(g)$. In particular, $A$ is reduced.
* For every $p \in Spec\_r(A)$ with corresponding maximal ideal $m\_p$, then the maximal ideal $\overline{m\_p}$ of $A\_{\mathfrak{m}\_p}$ is finitely generated, say by elements $f\_1,...,f\_n$, and the canonical map
$\mathbb{R}[t\_1,...,t\_n] / (t\_1,...,t\_n)^{r+1} \to A\_{\mathfrak{m}\_p} / \overline{m\_p}^{r+1}, t\_i \mapsto f\_i$
is an isomorphism for all $r \geq 0$.
* With the notation above, the canonical map $A/m\_p^{r+1} \to A\_{\mathfrak{m}\_p} / \overline{m\_p}^{r+1}$ is an isomorphism.
* The function $Spec\_r(A) \to \mathbb{N}, p \to \dim\_\mathbb{R} \mathfrak{m}\_p/{\mathfrak{m}\_p}^2$ is locally constant.


Are they sufficient [**no, see Michael's answer**]? Finally [**solved by Dmitri's answer**]:


* How can we characterize the algebras (at least within all the $C^{\infty}(M)$'s), that come from *compact* manifolds?


You might admit that "$Spec\_r(A)$ is compact with the Gelfand topolgy" is not a satisfactory answer ;-).


**Addendum**: 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 claim that this applies to my problem. But at least I invite you to think about it. The properties of the algebras above are just an approximation. Even if we add some of the conditions in the answers (such as $\cap\_{r} \overline{m}\_p^{r+1} \neq 0$), it would be a great surprise that the conditions are sufficient. But I'm not convinced of the contrary as soon someone provides a counterexample. It is fun trying to deduce some of the differential geometric theorems such as IFT from the properties above (if $A \to B$ is an isomorphism in one tangent space, then it is a local isomorphism). Perhaps a first step is to characterize the local rings $C^{\infty}\_p(\mathbb{R}^n)$.



How to classify the algebras C^∞(M)?
There is a strongly geometric characterization of those algebras which arise as $C^\infty(M)$ for $M$ compact and orientable, recently proved by Connes, see 
[here](http://arxiv.org/abs/0810.2088). This has come up on MO before, e.g. in Joel Fine's answer to this question:


[Algebraic description of compact smooth manifolds?](https://mathoverflow.net/questions/5344/algebraic-description-of-compact-smooth-manifolds)


Like the proofs of most major theorems in differential topology, Connes's approach invokes (a) Riemannian metrics, and (b) hard analysis. Spectral geometry is not my area, so this will be an amateurish explanation... 


If $M$ is a compact smooth manifold, $C^\infty(M)$ is represented faithfully on the Hilbert space $H$ of $L^2$ sections of any hermitian vector bundle $S$. If $D$ is a first-order differential operator acting (unboundedly) on $H$ then we can recover the projective $C^\infty(M)$-module of sections $C^\infty(M;S)$ within $H$ as $\bigcap\_{k>0}{dom(D^k)}$ (these domains will actually by Sobolev spaces, I believe). The algebraic counterpart of being first-order is that $[[D,f],g]=0$ for any $f,g \in C^\infty(M)$. 


$D$ has particularly nice properties when it's elliptic. There's no canonical elliptic operator over a smooth manifold until one chooses a Riemannian metric; there's then the signature operator $D=d+d^\ast$ acting on the complexified differential forms. This is an example of a Dirac operator (its square is a Laplacian - this is a condition on the symbol of the operator). As such, it's formally self-adjoint, Fredholm, and its (real) spectrum has known growth rate depending on $\dim(M)$. 


Connes (see Theorem 11.4) shows that a commutative $\mathbb{R}$-algebra $A$ arises as $C^\infty(M)$ for a smooth manifold structure on the space $M$ Gelfand-dual to $A$ provided that it's part of a "spectral triple" $(A,H,D)$ of the right kind. This means that $A$ should act on a Hilbert space $H$ carrying an unbounded symmetric operator $D$ satisfying various properties. I've hinted at some of these; the most sophisticated property is an "orientation" condition invoking a Hochschild cycle $c\in Z\_{\dim M}(A,A)$. This cycle is something like a volume form, and from its components Connes rebuilds local charts.


