Is the kernel of a map between finite dimensional vector bundles still of finite type?

Question

I'm not sure whether the level of this question is suitable for Mathoverflow.

Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow F$ be a bundle map.

We know that $\ker\phi$ is not necessarily a vector bundle if $\phi$ is not surjective. But is it true that $\ker\phi$ still of finite type? More precisely, is the following statement true?

For any point $x\in M$, there is an open neighborhood $U$ of $x$ and a finite dimensional smooth vector bundle $G$ on $U$, together with a surjective $\mathcal{C}^{\infty}(U)$-module map $\psi: G\rightarrow \ker \phi|_U$ (we treat $G$ and $\ker \phi$ as sheaves of $\mathcal{C}^{\infty}(U)$-modules.)

Answer 1

No. Namely, the rank of $\Phi:E\to F$ cannot drop locally (some minor, in local frames, has $\det\ne 0$). But it can increase locally. So the rank of $ker(\Phi)$ can drop locally. But the rank of $\Psi:G\to E$ cannot drop locally.