The Stacks project

28.19 Flat modules

On any ringed space $(X, \mathcal{O}_ X)$ we know what it means for an $\mathcal{O}_ X$-module to be flat (at a point), see Modules, Definition 17.17.1 (Definition 17.17.3). For quasi-coherent sheaves on an affine scheme this matches the notion defined in the algebra chapter.

slogan

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. $\square$

Comments (2)

Comment #2345 by Jonathan Gruner on

In Lemma 27.19.1, there seems to be a typo: “of if and only if”.

Also, in the text above the lemma, one could add that this is a statement about quasi-coherent sheaves: “For quasi-coherent sheaves on an affine scheme, this matches the notion defined in the algebra chapter.”

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