Lemma 74.11.21. The property $\mathcal{P}(f) =$“$f$ is a quasi-compact 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, Lemmas 67.12.1 and 67.8.8. 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 quasi-compact immersion. We have to show that $X \to Z$ is a closed immersion. The morphism $X' \to Z'$ is quasi-affine. Hence by Lemma 74.11.20 we see that $X$ is a scheme and $X \to Z$ is quasi-affine. It follows from Descent, Lemma 35.23.21 that $X \to Z$ is a quasi-compact immersion as desired. $\square$

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