Reconstructing a sheaf from its global sections

Question

Let $\mathcal{F}$ be a sheaf on a smooth manifold $M$ with the property that $\mathcal{F}(U)$ is a $C^{\infty}(U)$-module for every open subset $U\subseteq M$. I wonder if/when you can reconstruct $\mathcal{F}$ from its global sections $\mathcal{F}(M)$. Maybe you can define something like $\tilde{\mathcal{F}}(U):=C^{\infty}(U)\otimes_{C^{\infty}(M)}\mathcal{F}(M)$?

Answer 1

No. Let $M = (—1, 0) \cup (0, 1)$. Define $\mathcal{F}(U)$ to the the module $C^\infty(U \cap (0, 1))$. Define $\mathcal{G}(U) = \mathcal{F}(\{-x \mid x \in U\})$. Then we have $\mathcal{F}(M) = \mathcal{G}(M)$ as $C^\infty(M)$-modules, but the two sheaves aren’t isomorphic.