The Stacks project

Lemma 37.51.7. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $U \subset X$ be an open subscheme. Assume $f$ locally of finite type, $S$ locally Noetherian, $x$ a closed point of $X_ s$, and assume there exists a point $x' \in U$ with $x' \leadsto x$ and $f(x') \not= s$. Then there exists a closed subscheme $Z \subset X$ such that (a) $x \in Z$, (b) $f|_ Z : Z \to S$ is quasi-finite at $x$, and (c) there exists a $z \in Z$, $z \in U$, $z \leadsto x$ and $f(z) \not= s$.

Proof. This is a reformulation of Lemma 37.51.6. Namely, set $A = \mathcal{O}_{S, s}$ and $B = \mathcal{O}_{X, x}$. Denote $V \subset \mathop{\mathrm{Spec}}(B)$ the inverse image of $U$. The ring map $f^\sharp : A \to B$ is essentially of finite type. By assumption there exists at least one point of $V$ which does not map to the closed point of $\mathop{\mathrm{Spec}}(A)$. Hence all the assumptions of Lemma 37.51.6 hold and we obtain a prime $\mathfrak q \subset B$ which does not lie over $\mathfrak m_ A$ and such that $A \to B/\mathfrak q$ is the localization of a quasi-finite ring map. Let $z \in X$ be the image of the point $\mathfrak q$ under the canonical morphism $\mathop{\mathrm{Spec}}(B) \to X$. Set $Z = \overline{\{ z\} }$ with the induced reduced scheme structure. As $z \leadsto x$ we see that $x \in Z$ and $\mathcal{O}_{Z, x} = B/\mathfrak q$. By construction $Z \to S$ is quasi-finite at $x$. $\square$

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