Section module is finitely generated projective

Definition

Let $M$ be a compact 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.

Compactness guarantees the finite global construction used below.

Proof by a complementary bundle

A finite trivializing cover and a subordinate smooth partition of unity produce finitely many global sections that span every fiber. They define a surjective vector bundle morphism

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

After choosing a bundle metric, its kernel is a smooth vector subbundle and the orthogonal complement maps isomorphically onto $E$ . Thus there is a vector bundle $F$ with

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

Taking smooth sections gives

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

which proves both projectivity and finite generation Nestruev, Chapter 11.

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 Nestruev, Chapter 11.

The compact hypothesis makes the finite-cover proof immediate. Extensions to noncompact finite-dimensional manifolds use finite-covering-dimension or bundle-embedding results instead. If one replaces $C^\infty(M)$ by functions vanishing at infinity, the appropriate section module and projectivity statement must also be changed; this is a different Serre–Swan formulation Nestruev, Chapter 11.

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.”