The Stacks project

Lemma 37.44.3. Consider a commutative diagram of schemes

\[ \xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & Y \ar[ld]^ g \\ & S } \]

Let $s \in S$. Assume

  1. $X \to S$ is a proper morphism,

  2. $Y \to S$ is separated and locally of finite type, and

  3. the image of $X_ s \to Y_ s$ is finite.

Then there is an open subspace $U \subset S$ containing $s$ such that $X_ U \to Y_ U$ factors through a closed subscheme $Z \subset Y_ U$ finite over $U$.

Proof. Let $Z \subset Y$ be the scheme theoretic image of $h$, see Morphisms, Section 29.6. By Morphisms, Lemma 29.41.10 the morphism $X \to Z$ is surjective and $Z \to S$ is proper. Thus $X_ s \to Z_ s$ is surjective. We see that either (3) implies $Z_ s$ is finite. Hence $Z \to S$ is finite in an open neighbourhood of $s$ by Lemma 37.44.2. $\square$

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