Injectivity of a Morphism of Invertible Sheaves

Question

Let $\mathcal{F},\mathcal{G}$ be invertible sheaves on an integral scheme $X$ and $f:\mathcal{F} \rightarrow \mathcal{G}$ be a morphism. My question is simple. Is it the case that $f$ is injective if and only if $f$ is induces a nonzero map at generic point?

Answer 1

Your question comes down to: If $R$ is an integral domain and $a \in R$, is it true that $a : R \to R$ is injective iff $a \neq 0$ in $\mathrm{Quot}(R)$? Sure, and we may also add $a \neq 0$ in $R$ here.