Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?

Question

Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts to asking whether there exist finitely many smooth sections $s_1,...,s_m$ such that every $s\in \Gamma(E,M)$ can be written as $s=f_1 s_1+...+f_m s_m$ with smooth functions $f_1,...,f_m$.

Answer 1

If $M$ is finite-dimensional and second-countable, $\Gamma(E,M)$ is always finitely generated over $C^\infty(M)$. To see this, first note that $M$ has a finite covering $\{U_1,\dots,U_{k}\}$ consisting of open sets over which $E$ is trivial. (See this post for an explanation. Using the theory of topological covering dimension, you can show that it's possible to cover $M$ with $n+1$ such sets, where $n$ is the dimension of $M$.) For each $i=1,\dots,k$, let $(\sigma_{i1},\dots,\sigma_{im})$ be a smooth local frame for $E$ over $U_i$ (where $m$ is the rank of $E$). Let $\{\phi_1,\dots,\phi_{k}\}$ be a smooth partition of unity subordinate to $\{U_1,\dots,U_k\}$, and for each $i,j$, define $$ \tilde\sigma_{ij} = \left\{ \begin{matrix} \phi_i\sigma_{ij}, & \text{on }U_i,\\ 0 & \text{on }M\smallsetminus \text{supp }\phi_i. \end{matrix} \right. $$ For each $i$, let $V_i\subseteq M$ be the open set on which $\phi_i>0$. Because the $\phi_i$'s sum to $1$, $\{V_1,\dots,V_k\}$ is also an open cover of $M$. The $\tilde \sigma_{ij}$'s are smooth global sections of $E$, with the property that their restrictions to $V_i$ span $\Gamma(E,V_i)$ for each $i$. Let $\{\psi_1,\dots,\psi_k\}$ be a smooth partition of unity subordinate to $\{V_1,\dots,V_k\}$.

Now suppose $S$ is a global smooth section of $E$. Over each $V_i$, we can choose smooth functions $f_{i1},\dots,f_{im}\in C^\infty(V_i)$ such that $S|_{V_i} = f_{i1}\tilde \sigma_{i1}+\dots+f_{im}\tilde \sigma_{im}$. Therefore, globally we can write $$ S = \sum_{i=1}^k \psi_i(f_{i1}\tilde \sigma_{i1}+\dots+f_{im}\tilde \sigma_{im}) = \sum_{i,j} \tilde {f_{ij}} \tilde \sigma_{ij}, $$ where $\tilde {f_{ij}}$ is the globally defined smooth function $$ \tilde {f _{ij}} = \left\{ \begin{matrix} \psi_i f _{ij}, & \text{on }V_i,\\ 0 & \text{on }M\smallsetminus \text{supp }\psi_i. \end{matrix} \right. $$ Thus the sections $\tilde \sigma_{ij}$ generate $\Gamma(E,M)$.