Lemma 87.24.8 (0AQ7)—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 87.24.8 (cite)

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Lemma 87.24.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ . If $Y$ is locally Noetherian and $f$ locally of finite type, then $X$ is locally Noetherian.

Proof. Pick $\{ Y_ j \to Y\}$ and $\{ X_{ij} \to Y_ j \times _ Y X\}$ as in Lemma 87.24.6. Then it follows from Lemma 87.19.9 that each $X_{ij}$ is Noetherian. This proves the lemma. $\square$

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