Lemma 17.20.4 (0GMU)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.20: Flat morphisms of ringed spaces
Lemma 17.20.4 (cite)

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Lemma 17.20.4. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module flat over $Y$ . Then the functor

$\textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X),\quad \mathcal{G} \longmapsto f^*\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F}$

is exact.

Proof. This is true because $f^*\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} = f^{-1}\mathcal{G} \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{F}$ , the functor $f^{-1}$ is exact, and $\mathcal{F}$ is a flat $f^{-1}\mathcal{O}_ Y$ -module. $\square$

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