Lemma 67.34.8 (06LS)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.34: Morphisms and dimensions of fibres
Lemma 67.34.8 (cite)

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Lemma 67.34.8. Let $S$ be a scheme. Consider a cartesian diagram

\[ \xymatrix{ X \ar[d] & F \ar[l]^ p \ar[d] \\ Y & \mathop{\mathrm{Spec}}(k) \ar[l] } \]

where $X \to Y$ is a morphism of algebraic spaces over $S$ which is locally of finite type and where $k$ is a field over $S$. Let $z \in |F|$ be such that $\dim _ z(F) = 0$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism

\[ X \longrightarrow Y \]

is locally quasi-finite.

Proof. Let $X' \subset X$ be the open subspace over which $f$ is locally quasi-finite found in Lemma 67.34.7. Since the formation of $X'$ commutes with arbitrary base change we see that $z \in X' \times _ Y \mathop{\mathrm{Spec}}(k)$. Hence the lemma is clear. $\square$

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