Lemma 38.19.1. Let $f : X \to S$ be a morphism of finite type. Let $\mathcal{F}$ be a quasi-coherent sheaf of finite type on $X$. Assume $S$ is local with closed point $s$. Assume $\mathcal{F}$ is pure along $X_ s$ and that $\mathcal{F}$ is flat over $S$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ of quasi-coherent $\mathcal{O}_ X$-modules. Then the following are equivalent

the map on stalks $\varphi _ x$ is injective for all $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$, and

$\varphi $ is injective.

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