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The Stacks project

Lemma 67.8.1. Let S be a scheme. Let f : X \to Y be a representable morphism of algebraic spaces over S. The following are equivalent:

  1. f is quasi-compact (in the sense of Section 67.3), and

  2. for every quasi-compact algebraic space Z and any morphism Z \to Y the algebraic space Z \times _ Y X is quasi-compact.

Proof. Assume (1), and let Z \to Y be a morphism of algebraic spaces with Z quasi-compact. By Properties of Spaces, Definition 66.5.1 there exists a quasi-compact scheme U and a surjective étale morphism U \to Z. Since f is representable and quasi-compact we see by definition that U \times _ Y X is a scheme, and that U \times _ Y X \to U is quasi-compact. Hence U \times _ Y X is a quasi-compact scheme. The morphism U \times _ Y X \to Z \times _ Y X is étale and surjective (as the base change of the representable étale and surjective morphism U \to Z, see Section 67.3). Hence by definition Z \times _ Y X is quasi-compact.

Assume (2). Let Z \to Y be a morphism, where Z is a scheme. We have to show that p : Z \times _ Y X \to Z is quasi-compact. Let U \subset Z be affine open. Then p^{-1}(U) = U \times _ Y Z and the scheme U \times _ Y Z is quasi-compact by assumption (2). Hence p is quasi-compact, see Schemes, Section 26.19. \square


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