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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