Vector bundles and locally free sheaves

Statement

Let $M$ be a smooth manifold and $\mathbb F\in\{\mathbb R,\mathbb C\}$ . Taking local smooth sections defines an equivalence of categories

\mathbf{Vect}_{\mathbb F}(M) \;\simeq\; \mathbf{LocFree}\bigl(C^\infty_M(\mathbb F)\bigr), \qquad E\longmapsto\Gamma^\infty(-,E).

The source is the fixed-base category of finite-rank vector bundles, and the target consists of locally free sheaves of $C^\infty_M(\mathbb F)$ -modules whose rank is finite and locally constant. Neither side requires that rank to be globally bounded across the connected components of $M$ . On morphisms, a bundle map over $\operatorname{id}_M$ acts on local sections by postcomposition.

Reconstruction from a sheaf

Choose local isomorphisms $\mathcal E|_{U_i}\cong (C^\infty_M(\mathbb F)|_{U_i})^{r}$ . On overlaps, changes of frame are smooth maps

g_{ij}:U_i\cap U_j\longrightarrow GL_r(\mathbb F)

satisfying the cocycle identities. Gluing the trivial bundles $U_i\times\mathbb F^r$ with these transition functions produces a smooth vector bundle $E\to M$ . Its section sheaf is naturally isomorphic to $\mathcal E$ .

Likewise, a morphism of locally free sheaves is locally a matrix of smooth functions. These matrices glue to a unique bundle morphism over $\operatorname{id}_M$ , proving full faithfulness.

No compactness requirement

This equivalence is local and requires no compactness hypothesis. The global smooth Serre–Swan theorem over $C^\infty(M,\mathbb F)$ likewise needs no compactness when $M$ is a finite-dimensional Hausdorff second-countable manifold; its finite generation uses a finite-dimensional vector-bundle embedding theorem rather than only local freeness.

The two statements must therefore be separated:

every finite-rank bundle has a finite-rank locally free section sheaf on any smooth manifold, with no global bound on ranks across components;
for connected $M$ , its global section module is finitely generated projective; on disconnected $M$ , the same holds when the componentwise ranks are globally bounded.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: vector bundles, transition functions, and bundle maps.
Glen E. Bredon, Sheaf Theory, 2nd ed., Springer, 1997. DOI record. Relevant: sheaves of modules and locally free sheaves.