Collapsing a fibre of a proper family forces nearby ones to collapse too.

Lemma 76.35.3. Let $S$ be a scheme. Let

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

be a commutative diagram of morphism of algebraic spaces over $S$. Let $b \in B$ and let $\mathop{\mathrm{Spec}}(k) \to B$ be a morphism in the equivalence class of $b$. Assume

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

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

3. one of the following is true

1. the image of $|X_ k| \to |Y_ k|$ is finite,

2. the image of $|f|^{-1}(\{ b\} )$ in $|Y|$ is finite and $B$ is decent.

Then there is an open subspace $B' \subset B$ containing $b$ such that $X_{B'} \to Y_{B'}$ factors through a closed subspace $Z \subset Y_{B'}$ finite over $B'$.

Proof. Let $Z \subset Y$ be the scheme theoretic image of $h$, see Morphisms of Spaces, Section 67.16. By Morphisms of Spaces, Lemma 67.40.8 the morphism $X \to Z$ is surjective and $Z \to B$ is proper. Thus

$\{ x \in |X|\text{ lying over }b\} \to \{ z \in |Z|\text{ lying over }b\}$

and $|X_ k| \to |Z_ k|$ are surjective. We see that either (3)(a) or (3)(b) imply that $Z \to B$ is quasi-finite all points of $|Z|$ lying over $b$ by Decent Spaces, Lemma 68.18.10. Hence $Z \to B$ is finite in an open neighbourhood of $b$ by Lemma 76.35.2. $\square$

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