Lemma 38.10.8. Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ with image $s \in S$. If $X$ is flat at $x$ over $S$, then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

$V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$

which contains the unique point of $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ mapping to $x$ such that $V \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is flat and of finite presentation.

Proof. The question is local on $X$ and $S$, hence we may assume $X$ and $S$ are 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 Proposition 38.10.3 to $\mathbf{A}^ n_ S \to S$, the sheaf $\mathcal{F} = i_*\mathcal{O}_ X$ and the point $t$. We obtain an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme

$W \subset \mathbf{A}^ n_{\mathcal{O}_{S', s'}}$

such that the pullback of $i_*\mathcal{O}_ X$ is flat and of finite presentation. This means that $V := W \cap \big (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})\big )$ is the desired open subscheme. $\square$

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