Lemma 37.34.3 (05FD)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.34: Limit arguments
Lemma 37.34.3 (cite)

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Lemma 37.34.3. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module of finite presentation and flat over $S$ . Then we may choose a diagram as in Lemma 37.34.2 and sheaf $\mathcal{F}_0$ such that in addition $\mathcal{F}_0$ is flat over $S_0$ .

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$ , $X = \mathop{\mathrm{Spec}}(B)$ , and $\mathcal{F} = \widetilde{M}$ . As $f$ is of finite presentation we see that $B$ is of finite presentation as an $A$ -algebra, see Morphisms, Lemma 29.21.2. As $\mathcal{F}$ is of finite presentation over $\mathcal{O}_ X$ we see that $M$ is of finite presentation as a $B$ -module, see Properties, Lemma 28.16.2. As $\mathcal{F}$ is flat over $S$ we see that $M$ is flat over $A$ , see Morphisms, Lemma 29.25.2. Thus the lemma follows from Algebra, Lemma 10.168.1. $\square$

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