Section module is finitely generated projective

Statement

Let $M$ be a connected finite-dimensional Hausdorff second-countable smooth manifold, let $\mathbb F\in\{\mathbb R,\mathbb C\}$ , and let $E\to M$ be a smooth finite-rank $\mathbb F$ -vector bundle. The finite-projectivity theorem for the section module states that the module of smooth sections $\Gamma^\infty(M,E)$ is a finitely generated projective module over $C^\infty(M,\mathbb F)$ . Equivalently, there are an integer $N$ , another $C^\infty(M,\mathbb F)$ -module $Q$ , and an isomorphism

\Gamma^\infty(M,E)\oplus Q\cong C^\infty(M,\mathbb F)^N.

No compactness hypothesis is required.

Proof by a complementary bundle

The finite-dimensional vector-bundle embedding theorem gives a finite-rank bundle $F$ and an isomorphism

E\oplus F\cong M\times\mathbb F^N.

Equivalently, the finite-dimensional global-generator theorem produces finitely many global sections that span every fiber. This conclusion does not require a finite trivializing cover; it follows from finite covering dimension together with a locally finite trivialization and a smooth partition of unity. The generators define a surjective vector bundle morphism

M\times\mathbb F^N\longrightarrow E.

Its kernel is a smooth vector subbundle. After choosing a bundle metric, the orthogonal complement maps isomorphically onto $E$ , recovering the displayed complement. Taking smooth sections gives

\Gamma(E)\oplus\Gamma(F)\cong C^\infty(M,\mathbb F)^N,

which proves both projectivity and finite generation.

Idempotent form

The splitting determines a smooth idempotent matrix

p\in M_N(C^\infty(M,\mathbb F)), \qquad p^2=p,

such that

\Gamma^\infty(M,E)\cong p\,C^\infty(M,\mathbb F)^N.

Conversely, the pointwise images of such an idempotent form a smooth bundle; this is the Serre–Swan idempotent construction. The theorem in the core is therefore one direction of the smooth Serre–Swan equivalence.

Examples and scope

For the trivial rank- $r$ bundle, $\Gamma^\infty(M,M\times\mathbb F^r)\cong C^\infty(M,\mathbb F)^r$ , which is free. The Möbius line bundle over $S^1$ gives a projective module that is not free: projectivity records the existence of a complementary bundle, whereas freeness would give a global frame and trivialize the bundle.

For disconnected $M$ , the conclusion remains valid when the ranks of $E$ on its connected components are globally bounded. Without that bounded-rank condition, a componentwise finite-rank bundle can have a section module that is not finitely generated. If one replaces $C^\infty(M,\mathbb F)$ by functions vanishing at infinity, the appropriate section module and projectivity statement must also be changed; that is a different Serre–Swan formulation.

References

Richard G. Swan, “Vector Bundles and Projective Modules,” Transactions of the American Mathematical Society 105 (1962), 264–277. DOI record. Relevant: Theorem 1 and the compact-Hausdorff continuous model for section modules as finite projective modules.
Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: Chapter 11, “Vector Bundles and Projective Modules.”