Module of smooth sections

Let $\mathbb F\in\{\mathbb R,\mathbb C\}$ , and let $E\to M$ be a smooth $\mathbb F$ -vector bundle over a smooth manifold. The module of smooth sections is

\Gamma^\infty(M,E)=\{s:M\to E\mid s\text{ is a smooth section of }E\}.

For $s,t\in\Gamma^\infty(M,E)$ and $f\in C^\infty(M,\mathbb F)$ , its operations are defined fiberwise by

(s+t)(x)=s(x)+t(x), \qquad (fs)(x)=f(x)s(x).

These operations make $\Gamma^\infty(M,E)$ a module over the commutative algebra $C^\infty(M,\mathbb F)$ . This module is also written $\Gamma(E)$ when smoothness, the scalar field, and the base are understood.

Local structure

Over a trivializing open set $U$ , a local frame identifies

\Gamma^\infty(U,E|_U)\cong C^\infty(U,\mathbb F)^r.

More precisely, these modules over varying open sets form the sheaf of smooth sections, which is locally free of rank $r=\operatorname{rank}E$ over $C^\infty_M(-,\mathbb F)$ . The single global module $\Gamma^\infty(M,E)$ is not itself what “locally free” refers to. Globally it need not possess a basis: a global module basis would be a global frame and would trivialize $E$ .

Evaluation at $x$ gives a surjective linear map $\Gamma^\infty(M,E)\to E_x$ . Its kernel consists of sections vanishing at $x$ , and the fiber can be recovered algebraically as

E_x\cong \Gamma^\infty(M,E)/\mathfrak m_x\Gamma^\infty(M,E),

where $\mathfrak m_x$ is the ideal of smooth functions vanishing at $x$ .

Functoriality

A smooth $\mathbb F$ -linear vector bundle morphism $\Phi:E\to E'$ over the identity of $M$ induces a $C^\infty(M,\mathbb F)$ -linear map

\Gamma(\Phi):\Gamma^\infty(M,E)\to\Gamma^\infty(M,E'), \qquad s\mapsto\Phi\circ s.

Conversely, $C^\infty(M,\mathbb F)$ -linear maps between section modules arise from unique smooth $\mathbb F$ -linear bundle morphisms under the standard finite-rank hypotheses. This correspondence is one part of the smooth Serre–Swan viewpoint.

The requirement that $\Phi$ cover $\operatorname{id}_M$ is essential: it places $E$ and $F$ in the same fixed-base bundle category and makes both section spaces modules over the same ring. A morphism covering $f:M\to N$ does not give the displayed map by postcomposition.

Relation to Serre–Swan

If $M$ is a connected finite-dimensional Hausdorff second-countable manifold and $E$ has finite rank, then $\Gamma^\infty(M,E)$ is a finitely generated projective module over $C^\infty(M,\mathbb F)$ , and every such module is the section module of a smooth $\mathbb F$ -vector bundle. This smooth Serre–Swan statement requires no compactness hypothesis; finite dimensionality supplies a finite-rank complementary bundle. On a disconnected base, one requires the ranks on components to be globally bounded. The original compact Hausdorff theorem over $C(X)$ is a distinct continuous formulation.

Taking sections over every open set gives the equivalence between vector bundles and finite-rank locally free $C^\infty_M(-,\mathbb F)$ -module sheaves.

References

R. 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 equivalence.
J. Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: Chapter 11, vector bundles and projective modules over smooth-function algebras.
L. W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: Chapter 1, smooth vector bundles and section modules.