Lemma 74.11.3. Let S be a scheme. The property \mathcal{P}(f) =“f is universally closed” 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 67.9.5. 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 universally closed. We have to show that f is universally closed. To see this, using Morphisms of Spaces, Lemma 67.9.5 again, it is enough to show that for every affine scheme Y and morphism Y \to Z the map |Y \times _ Z X| \to |Y| is closed. Consider the cube (74.11.1.1). The assumption that f' is universally closed implies that |Y \times _ Z Z' \times _ Z X| \to |Y \times _ Z Z'| is closed. As Y \times _ Z Z' \to Y is quasi-compact, surjective, and flat as a base change of Z' \to Z we see the map |Y \times _ Z Z'| \to |Y| is submersive, see Morphisms, Lemma 29.25.12. Moreover the map
is surjective, see Properties of Spaces, Lemma 66.4.3. It follows by elementary topology that |Y \times _ Z X| \to |Y| is closed. \square
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