The Stacks project

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

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

  2. $\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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