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.
Proof. Let \mathcal{K} = \mathop{\mathrm{Ker}}(\varphi ). Our goal is to prove that \mathcal{K} = 0. In order to do this it suffices to prove that \text{WeakAss}_ X(\mathcal{K}) = \emptyset , see Divisors, Lemma 31.5.5. We have \text{WeakAss}_ X(\mathcal{K}) \subset \text{WeakAss}_ X(\mathcal{F}), see Divisors, Lemma 31.5.4. As \mathcal{F} is flat we see from Lemma 38.13.5 that \text{WeakAss}_ X(\mathcal{F}) \subset \text{Ass}_{X/S}(\mathcal{F}). By purity any point x' of \text{Ass}_{X/S}(\mathcal{F}) is a generalization of a point of X_ s, and hence is the specialization of a point x \in \text{Ass}_{X_ s}(\mathcal{F}_ s), by Lemma 38.18.1. Hence the injectivity of \varphi _ x implies the injectivity of \varphi _{x'}, whence \mathcal{K}_{x'} = 0. \square
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