Lemma 38.12.6 (05IE)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.12: Flat finitely presented modules
Lemma 38.12.6 (cite)

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Lemma 38.12.6. Let $f : X \to S$ be locally of finite presentation. Let $x \in X$ with image $s \in S$ . If $f$ is flat at $x$ over $S$ , then there exists a commutative diagram of pointed schemes

$\xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (S, s) & (S', s') \ar[l] }$

whose horizontal arrows are elementary étale neighbourhoods such that $X'$ , $S'$ are affine and such that $\Gamma (X', \mathcal{O}_{X'})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$ -module.

Proof. This is a special case of Proposition 38.12.4. $\square$

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