Lemma 67.45.1 (03ZO)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.45: Integral and finite morphisms
Lemma 67.45.1 (cite)

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Lemma 67.45.1. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$ . Then $f$ is integral, resp. finite (in the sense of Section 67.3), if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times _ Y Z$ is affine and integral, resp. finite, over $Z$ .

Proof. This follows directly from the definition of an integral (resp. finite) morphism of schemes (Morphisms, Definition 29.44.1). $\square$

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