Let $X$ be a sufficiently nice scheme and $\mathcal F$ a locally free sheaf of finite rank, i.e., a vector bundle on $X$.
Let $x$ be point of $X$ and $k(x)$ it's residue field.
Let $f:\mathcal F \to \mathcal F$ be a morphism of sheaves; then $f$ induces a map
$f_x: \mathcal F_x \rightarrow F_x ,$
and a map
$\bar{f_x}: \mathcal F_x \otimes k(x) \rightarrow \mathcal F_x \otimes k(x) .$
Question:
If I know that $f_x$ is zero, then can I conclude that $f$ is zero in a sufficiently small neighborhood of $x$? And, still more important for me, if $\bar{f}_x$ is zero, can I conclude that $f$ is zero in a sufficiently small neighorhood of $x$?
Perhaps one may also assume $\mathcal F$ just as coherent, I'm not sure.