Lemma 79.12.4 (04PH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.12: The finite part of a morphism
Lemma 79.12.4 (cite)

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Lemma 79.12.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $T$ be an algebraic space over $S$ , and let $(a, Z)$ be a pair as in 79.12.0.1. If $f$ is separated, then $Z$ is closed in $T \times _ Y X$ .

Proof. A finite morphism of algebraic spaces is universally closed by Morphisms of Spaces, Lemma 67.45.9. Since $f$ is separated so is the morphism $T \times _ Y X \to T$ , see Morphisms of Spaces, Lemma 67.4.4. Thus the closedness of $Z$ follows from Morphisms of Spaces, Lemma 67.40.6. $\square$

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