Lemma 21.17.15 (08FG)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.17: Flat resolutions
Lemma 21.17.15 (cite)

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Lemma 21.17.15. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$ -module. The following are equivalent

$\mathcal{F}$ is a flat $\mathcal{O}$ -module, and
$\text{Tor}_1^\mathcal {O}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}$ -module $\mathcal{G}$ .

Proof. If $\mathcal{F}$ is flat, then $\mathcal{F} \otimes _\mathcal {O} -$ is an exact functor and the satellites vanish. Conversely assume (2) holds. Then if $\mathcal{G} \to \mathcal{H}$ is injective with cokernel $\mathcal{Q}$ , the long exact sequence of $\text{Tor}$ shows that the kernel of $\mathcal{F} \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{H}$ is a quotient of $\text{Tor}_1^\mathcal {O}(\mathcal{F}, \mathcal{Q})$ which is zero by assumption. Hence $\mathcal{F}$ is flat. $\square$

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