Lemma 74.11.17. The property $\mathcal{P}(f) =$“$f$ is a closed immersion” is fpqc local on the base.

**Proof.**
We will use Lemma 74.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 67.12.1. Consider a cartesian diagram

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 74.11.16 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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