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Let $E$ and $F$ be complex vector bundles on a smooth manifold $M$. I'm trying to prove that the $C^{\infty}(M)-$module morphisms $L: \Gamma(E) \rightarrow \Gamma(F)$ are in correspondence with the bundle morphisms $\tilde{L}: E \rightarrow F$. But I can't see how to construct such a bundle morphism from a $C^{\infty}(M)-$module morphism $L$.

My attempt so far: Given $e \in E$, let's choose a local smooth section $s: U \subset M \rightarrow E$ defined on a neighbourhood of $\pi_E(e)$ because $\pi_E$ is a submersion. Also, let's choose a bump function $\chi \in C^{\infty}(M)$ with compact support, and with the property that there exists a neighbourhood of $\pi_E(e)$ on which $\chi \equiv 1$. Then, $\chi s$ has compact support, and we can extend it trivially to all of $M$. Let's define $$\tilde{L}(e) = L(\chi s)(\pi_E(e)).$$

I proved that this definition is good after fixing $s$, but I couldn't prove that it is independent of the choice of s.

Thank you in advance for any help.

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  • $\begingroup$ Hint: use a local frame. $\endgroup$
    – Kajelad
    Jun 7, 2023 at 23:22

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