The Stacks project

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

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

of algebraic spaces over $S$ where $Z' \to Z$ is a surjective flat morphism of affine schemes, and $X' \to Z'$ is a 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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