Is there a "Hilbert syzygy theorem" for smooth manifolds? Or: does every finitely generated $C^\infty$ module have a finite-length resolution in vector bundles?

Question

Let $X$ be a real smooth manifold, and $M$ a locally-finitely-generated sheaf of $\mathcal C^\infty(X)$-modules. (If $X$ is not compact, I will also insist that there be a global bound on the number of generators I might need in different regions; maybe this is part of the usual meaning of the words "locally finitely generated".)

I would like to find finite-dimensional vector bundles $E_1,\dots,E_n$ over $X$ and maps of $\mathcal C^\infty$-modules $$ 0 \to \Gamma(E_n) \to \dots \to \Gamma(E_1) \to M \to 0$$ so that the sequence is exact. Can I always do this? And is there an explicit bound on the number of vector bundles needed, e.g. $n = \dim X$ or $n = \dim X+1$?

Answer 1

No. Let $X$ be $\mathbb R$. In the ring $C^{\infty}(X)$ let $I$ be the ideal of all functions vanishing to infinite order at $0$. The module $C^{\infty}(X)/I$ does not have a finite resolution by finitely generated projective modules.

Edit:

Still no if you want the finitely generated module to be contained in a finitely generated projective module. For the same $X$ pick a function $f$ such $f(x)$ vanishes precisely when $x<0$. let $J$ be the ideal generated by $f$. The module $J$ does not have a finite resolution by finitely generated projective modules.

For both of these examples, the method I have in mind is this: If a module $M$ has a finite projective resolution $P_\bullet$ then for every point in $p\in X$ the alternating sum of the $k_p$ vector space dimension of $Tor_n(M,k_p)$ is independent of $p$ because it's the alternating sum of the rank of $P_n$. I believe that in the first example this Euler number comes out to be $1$ if $p$ is the origin and otherwise $0$, and in the second it's $1$ if $p> 0$ and $0$ if $p< 0$.