The Stacks project

Lemma 73.11.17. The property $\mathcal{P}(f) =$“$f$ is a closed immersion” is fpqc local on the base.

Proof. We will use Lemma 73.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 66.12.1. Consider a cartesian diagram

\[ \xymatrix{ X' \ar[r] \ar[d] & X \ar[d] \\ Z' \ar[r] & Z } \]

of algebraic spaces over $S$ where $Z' \to Z$ is a surjective flat morphism of affine schemes, and $X' \to Z'$ is a closed immersion. We have to show that $X \to Z$ is a closed immersion. The morphism $X' \to Z'$ is affine. Hence by Lemma 73.11.16 we see that $X$ is a scheme and $X \to Z$ is affine. It follows from Descent, Lemma 35.23.19 that $X \to Z$ is a closed immersion as desired. $\square$


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