Lemma 38.2.4 (05BA)—The Stacks project

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Part 2: Schemes
Chapter 38: More on Flatness
Section 38.2: Lemmas on étale localization
Lemma 38.2.4 (cite)

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Lemma 38.2.4. Let $T \to S$ be an étale morphism. Let $t \in T$ with image $s \in S$ . Let $M$ be a $\mathcal{O}_{T, t}$ -module. Then

$M\text{ flat over }\mathcal{O}_{S, s} \Leftrightarrow M\text{ flat over }\mathcal{O}_{T, t}.$

Proof. We may replace $S$ by an affine neighbourhood of $s$ and after that $T$ by an affine neighbourhood of $t$ . Set $\mathcal{F} = (\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to T)_*\widetilde M$ . This is a quasi-coherent sheaf (see Schemes, Lemma 26.24.1 or argue directly) on $T$ whose stalk at $t$ is $M$ (details omitted). Apply Lemma 38.2.3. $\square$

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