Vector bundles and projective $C^{\infty}$-modules

Question

There is the following theorem:

Let $M$ be a smooth manifold (Hausdorff, paracompact etc.). Then there is a bijection

{vector bundles over M}$/\cong \ \longleftrightarrow $ {projective $C^{\infty}(M)$-modules}$/\cong$.

There is an easy (functorial) way to get from vector bundles to projective modules over $C^{\infty}(M)$, by taking global sections.

My question comes from a similar result in Algebraic Geometry:

Let $X$ be a scheme. Then there exists an equivalence of categories:

{vector bundles over $X$} $\cong$ {locally free sheaves on $X$}

where the functors are given by taking the sheaf of sections (which is similar to global sections of the vector bundle in the setting of manifolds) and taking the global spectrum of the symmetric algebra of the locally free sheaf in the other direction.

So in the setting of schemes, the connection seems to be much nicer and I'm asking myself, whether there is a similar procedure in the world of manifolds that gives a functor from projective $C^{\infty}(M)$-modules (or locally free $C^{\infty}_M$-modules) to vector bundles over $M$.

(I'm sorry for the vague formulation, but I'm not really sure what I want. It worries me that the construction of the vector bundle associated to a projective $C^{\infty}(M)$-module seems non-canonical)

Answer 1

Given a projective module $P$ on a commutative ring $A$, you construct a vector bundle on the affine scheme $Spec A$ by taking the spectrum of the symmetric algebra of $P^\vee$: $$SpecSym^*_AP^\vee\to Spec A.$$

For a compact smooth manifold $M$ you can do a similar construction: given a projective $\mathcal{C}^\infty(M)$-module $P$, you get a vector bundle by taking the real spectrum (with real topology) $$Spec_\mathbb{R}Sym^*_{\mathcal{C}^\infty(M)}P^\vee \to Spec_\mathbb{R}\mathcal{C}^\infty(M)\simeq M$$ of the symmetric algebra of $P^\vee$. This construction defines a quasi-inverse functor to the one which takes global sections.

Details can be found e.g. in section 5.3 of this book.

Notice that you have a "local" statement in the smooth world also: there is an equivalence of categories between vector bundles on $M$ and locally free sheaves of $\mathcal{C}^\infty_M$-modules (and in this case compactness is not needed).