Lemma 17.17.7 (05NJ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.17: Flat modules
Lemma 17.17.7 (cite)

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Lemma 17.17.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let

\[ 0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0 \]

be a short exact sequence of $\mathcal{O}_ X$-modules. Assume $\mathcal{F}$ is flat. Then for any $\mathcal{O}_ X$-module $\mathcal{G}$ the sequence

\[ 0 \to \mathcal{F}'' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F}' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \to 0 \]

is exact.

Proof. Using that $\mathcal{F}_ x$ is a flat $\mathcal{O}_{X, x}$-module for every $x \in X$ and that exactness can be checked on stalks, this follows from Algebra, Lemma 10.39.12. $\square$

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