Let $M$ be a smooth manifold, and $\mathfrak{X}(M)$ be the associated $C^\infty(M)$-module of vector fields on $M$.
We say that $P\subseteq \mathfrak{X}(M)$ is finitely generated if there exists finitely many elements $X_1,\cdots,X_n$ in $P$ such that, $$P=\left<X_1,\cdots,X_n\right>.$$ We say that $P\subseteq \mathfrak{X}(M)$ is locally finitely generated if there exists an open cover $\{U_\alpha\}$ of $M$ such that $P|_{U_\alpha}$ is finitely generated for each $\alpha\in \Lambda$.
I am thinking of the following question:
Suppose $M$ is a compact manifold. Is it true that any locally finitely generated $C^\infty(M)$ submodule of $\mathfrak{X}(M)$ us a finitely generated submodule?
If I am not mistaken, this is what is mentioned in some result.
Let $P$ be a locally finitely generated $C^\infty(M)$-submodule of $\mathfrak{X}(M)$. So, there is an open cover $\{U_\alpha\}$ of $M$ such that $P|_{U_\alpha}$ is finitely generated for each $\alpha$. Suppose that $M$ is compact, then, this open cover $\{U_\alpha\}$ will have a finite sub cover, say $\{U,V\}$. Just for simplicity, assume $P|_U=\left<X\right>$ and $P|_V=\left<Y\right>$. Now, I am trying to see if we can combine these vector fields $X\in \Gamma(U,TU)$ and $Y\in \Gamma(V,TV)$ to get a (finite collection of) vector field(s) that generate $P$.
This may be straight forward, but I am not able to get it now.
The key trick is to use a partition of unity. Suppose you have a finite open cover $M=U_1\cup\dots\cup U_n$ such that $P|_{U_i}$ is finitely generated for each $i$. Take a partition of unity $1=\sum_i\varphi_i$ where each $\varphi_i\in C^\infty(M)$ is supported on $U_i$. Now if $P|_{U_i}$ is generated by the restrictions of finitely many vector fields $X_{ij}\in P$, I claim that $P$ is generated by all the vector fields $X_{ij}$. Indeed, if $X\in P$, then for each $i$ there are $f_{ij}\in C^\infty(U_i)$ such that $X=\sum_j f_{ij}X_{ij}$ on $U_i$. On all of $M$, we then have $X=\sum_i\varphi_iX=\sum_i\sum_j\varphi_i f_{ij}X_{ij}$, where this expression makes sense on all of $M$ since $\varphi_i$ is supported on $U_i$ so $\varphi_if_{ij}$ can be considered as a smooth function on all of $M$. This writes $X$ as a $C^{\infty}(M)$-linear combination of the $X_{ij}$.