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Let Let $E \rightarrow X$ be a vector bundle on a manifold $X$. Let $\cal E$ be the sheaf of sections of $E$. Let $\cal F$ be a subsheaf of $\cal E$, and let $F$ be the etale space of $\cal F$. What is an example that the map $F \rightarrow E$ might not be an injection on all fibers?

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Take $E$ to be the trivial bundle $X \times \mathbb{R}$. Let $f: X \to \mathbb{R}$ to be a function vanishing only at $x_0$. Then multiplication by the global function $f$ defines a morphism $E \to E$ which is injective as a morphism of sheaves (because it is injective on $E(U)$ for any open set $U$), but the map on fibers at $x_0$ is not injective.

(To avoid uninteresting cases, assume $x_0$ is not an isolated point of $X$.)

The problem is that for an injection of sheaves $E \to F$, the map on stalks $E_x \to F_x$ is always injective. The map on fibers $E(x) \to F(x)$ obtained by tensoring the map on stalks with the residue field is generally not, because tensoring is not an exact functor. If the map on stalks is a split injection, then the map on fibers will be an injection still.

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