The Stacks project

Proof. During the proof we may replace $X$ by an open neighbourhood of $x$ and we may replace $S$ by an elementary étale neighbourhood of $s$. Hence, by openness of flatness (see More on Morphisms, Theorem 37.15.1) we may assume that $\mathcal{F}$ is flat over $S$. We may assume $S$ and $X$ are affine. After shrinking $X$ some more we may assume that any point of $\text{Ass}_{X_ s}(\mathcal{F}_ s)$ is a generalization of $x$. This property is preserved on replacing $(S, s)$ by an elementary étale neighbourhood. Hence we may apply Lemma 38.12.5 to arrive at the situation where there exists a diagram

of schemes affine and of finite presentation over $S$, where $g$ is étale, $X_ s \subset g(X')$, and with $\Gamma (X', g^*\mathcal{F})$ a projective $\Gamma (S, \mathcal{O}_ S)$-module. Note that in this case $g^*\mathcal{F}$ is universally pure over $S$, see Lemma 38.17.4.

Let $U \subset g(X')$ be an affine open neighbourhood of $x$. We claim that $\mathcal{F}|_ U$ is pure along $U_ s$. If we prove this, then the lemma follows because $\mathcal{F}|_ U$ will be pure relative to $S$ after shrinking $S$, see Lemma 38.18.6, whereupon the projectivity follows from Proposition 38.19.2. To prove the claim we have to show, after replacing $(S, s)$ by an arbitrary elementary étale neighbourhood, that any point $\xi $ of $\text{Ass}_{U/S}(\mathcal{F}|_ U)$ lying over some $s' \in S$, $s' \leadsto s$ specializes to a point of $U_ s$. Since $U \subset g(X')$ we can find a $\xi ' \in X'$ with $g(\xi ') = \xi $. Because $g^*\mathcal{F}$ is pure over $S$, using Lemma 38.18.1, we see there exists a specialization $\xi ' \leadsto x'$ with $x' \in \text{Ass}_{X'_ s}(g^*\mathcal{F}_ s)$. Then $g(x') \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ (see for example Lemma 38.2.8 applied to the étale morphism $X'_ s \to X_ s$ of Noetherian schemes) and hence $g(x') \leadsto x$ by our choice of $X$ above! Since $x \in U$ we conclude that $g(x') \in U$. Thus $\xi = g(\xi ') \leadsto g(x') \in U_ s$ as desired. $\square$