Lemma 38.19.4. Let f : X \to S be a morphism which is locally of finite type. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module which is of finite type. Let x \in X with s = f(x) \in S. If \mathcal{F} is flat at x over S there exists an affine elementary étale neighbourhood (S', s') \to (S, s) and an affine open U' \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) which contains x' = (x, s') such that \Gamma (U', \mathcal{F}|_{U'}) is a free \mathcal{O}_{S', s'}-module.
Proof. The question is Zariski local on X and S. Hence we may assume that X and S are affine. Then we can find a closed immersion i : X \to \mathbf{A}^ n_ S over S. It is clear that it suffices to prove the lemma for the sheaf i_*\mathcal{F} on \mathbf{A}^ n_ S and the point i(x). In this way we reduce to the case where X \to S is of finite presentation. After replacing S by \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) and X by an open of X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) we may assume that \mathcal{F} is of finite presentation, see Proposition 38.10.3. In this case we may appeal to Lemma 38.19.3 and Algebra, Theorem 10.85.4 to conclude. \square
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