Lemma 28.19.1 (05P0)—The Stacks project

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Flatness is the same for modules and sheaves.

Lemma 28.19.1. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme. Let $\mathcal{F} = \widetilde{M}$ for some $R$-module $M$. The quasi-coherent sheaf $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module if and only if $M$ is a flat $R$-module.

Proof. Flatness of $\mathcal{F}$ may be checked on the stalks, see Modules, Lemma 17.17.2. The same is true in the case of modules over a ring, see Algebra, Lemma 10.39.18. And since $\mathcal{F}_ x = M_{\mathfrak p}$ if $x$ corresponds to $\mathfrak p$ the lemma is true. $\square$

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Comment #1224 by David Corwin on December 28, 2014 at 03:33

Suggested slogan: Flatness is the same for modules and sheaves

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