This continues my question about smooth Gelfand-duality. In the book

Juan A. Navarro González & Juan B. Sancho de Salas, C

^{∞}-Differentiable Spaces, LNM 1824

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 $C^{\infty}_p(\mathbb{R}^n)$.