The Stacks project

Lemma 37.48.9. Suppose that $f : X \to S$ is locally of finite type, $S$ locally Noetherian, $x \in X$ a closed point of its fibre $X_ s$, and $U \subset X$ an open subscheme such that $U \cap X_ s = \emptyset $ and $x \in \overline{U}$, then the conclusions of Lemma 37.48.7 hold.

Proof. Namely, we can reduce this to the cited lemma as follows: First we replace $X$ and $S$ by affine neighbourhoods of $x$ and $s$. Then $X$ is Noetherian, in particular $U$ is quasi-compact (see Morphisms, Lemma 29.15.6 and Topology, Lemmas 5.9.2 and 5.12.13). Hence there exists a specialization $x' \leadsto x$ with $x' \in U$ (see Morphisms, Lemma 29.6.5). Note that $f(x') \not= s$. Thus we see all hypotheses of the lemma are satisfied and we win. $\square$


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