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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