Surjectivity of map of étale sheaves

Question

Let $F\to G$ be a map of étale sheaves on a scheme $X$.

Suppose that for any closed point $x\in X$, the map between stalks

$$F_x\to G_x$$ is surjective.

Note that I am not forming stalks at the geometric points lying over $x$, but at $x$ itself.

Is $F\to G$ surjective?

Answer 1

If I am understanding your question correctly, the answer is no...

Consider the inclusion $\mu_2\to \mu_4$ over $\mathop{\rm spec} {\mathbb Q}$. There is only one point, and the stalks are $\mu_2 ( {\mathbb Q } ) = \mu_4 ( {\mathbb Q } )=\{\pm 1\}$. But the map is not surjective. (Here, $\mu_k(A)$ denotes the $k$th roots of unity in $A$.)