Lemma 37.23.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

$f$ is locally of finite presentation,

$f$ is flat at $x$, and

$\mathcal{O}_{X_ s, x}$ has $\text{depth} \geq 1$.

Then there exists an affine open neighbourhood $U \subset X$ of $x$ and an effective Cartier divisor $D \subset U$ containing $x$ such that $D \to S$ is flat and of finite presentation.

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