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.
Flatness is the same for modules and sheaves.
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 (1)
Comment #1224 by David Corwin on
There are also: