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