Pull-back of injective morphism of locally free sheaves is injective? Let $i:X \hookrightarrow \mathbb{P}^n$ be a smooth projective variety. Let $f:\mathcal{F}_1 \to \mathcal{F}_2$ be an injective morphism of $\mathcal{O}_{\mathbb{P}^n}$-modules that are locally free. Is the induced morphism $i^*:i^*\mathcal{F}_1 \to i^*\mathcal{F}_2$ injective?

No, this is not true.

Denote by $S=k[X_0,...,X_n]$, $X$ a hypersurface in $\mathbb{P}^n$ defined by an equation $F$ of degree $d$. Then there is an injective morphism $S(-d) \to S$ given by multiplication by $F$. But the induced morphism $(S/(F))(-d) \to S/(F)$ is not injective.

Applying the associated coherent sheaf functor to the modules give a counterexample.