Lemma 38.10.10. Let f : X \to S be a morphism which is locally of finite type. Let x \in X with image s \in S. If f is flat at x over S, then \mathcal{O}_{X, x} is essentially of finite presentation over \mathcal{O}_{S, s}.
Proof. We may assume X and S affine. Write X = \mathop{\mathrm{Spec}}(B), S = \mathop{\mathrm{Spec}}(A) and write B = A[x_1, \ldots , x_ n]/I. In other words we obtain a closed immersion i : X \to \mathbf{A}^ n_ S. Denote t = i(x) \in \mathbf{A}^ n_ S. We may apply Lemma 38.10.9 to \mathbf{A}^ n_ S \to S, the sheaf \mathcal{F} = i_*\mathcal{O}_ X and the point t. We conclude that \mathcal{O}_{X, x} is of finite presentation over \mathcal{O}_{\mathbf{A}^ n_ S, t} which implies what we want. \square
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