Lemma 87.24.2 (0AJJ)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.24: Morphisms of finite type
Lemma 87.24.2 (cite)

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Lemma 87.24.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ . The following are equivalent

$f$ is of finite type,
$f$ is representable by algebraic spaces and is of finite type in the sense of Bootstrap, Definition 80.4.1.

Proof. This follows from Bootstrap, Lemma 80.4.5, the implication “quasi-compact $+$ locally of finite type $\Rightarrow$ finite type” for morphisms of algebraic spaces, and Lemma 87.17.5. $\square$

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