Lemma 38.4.3 (05H6)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.4: One step dévissage
Lemma 38.4.3 (cite)

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Lemma 38.4.3. Let $f : X \to S$ be morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$ -module. Let $x \in X$ with image $s = f(x)$ in $S$ . Then there exists a commutative diagram of pointed schemes

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

such that $(S', s') \to (S, s)$ and $(X', x') \to (X, x)$ are elementary étale neighbourhoods, and such that $g^*\mathcal{F}/X'/S'$ has a one step dévissage at $x'$ .

Proof. This is immediate from Definition 38.4.2 and Lemma 38.3.2. $\square$

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