Lemma 74.11.1. Let S be a scheme. The property \mathcal{P}(f) =“f is quasi-compact” is fpqc local on the base on algebraic spaces over S.
Proof. We will use Lemma 74.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 67.8.8. Let Z' \to Z be a surjective flat morphism of affine schemes over S. Let f : X \to Z be a morphism of algebraic spaces, and assume that the base change f' : Z' \times _ Z X \to Z' is quasi-compact. We have to show that f is quasi-compact. To see this, using Morphisms of Spaces, Lemma 67.8.8 again, it is enough to show that for every affine scheme Y and morphism Y \to Z the fibre product Y \times _ Z X is quasi-compact. Here is a picture:
Note that all squares are cartesian and the bottom square consists of affine schemes. The assumption that f' is quasi-compact combined with the fact that Y \times _ Z Z' is affine implies that Y \times _ Z Z' \times _ Z X is quasi-compact. Since
is surjective as a base change of Z' \to Z we conclude that Y \times _ Z X is quasi-compact, see Morphisms of Spaces, Lemma 67.8.6. This finishes the proof. \square
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