The Stacks project

Lemma 37.22.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume

  1. $f$ is locally of finite presentation,

  2. $f$ is flat at $x$, and

  3. $\mathcal{O}_{X_ s, x}$ 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.

Proof. Pick any $h \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$ which maps to a nonzerodivisor in $\mathcal{O}_{X_ s, x}$ and apply Lemma 37.22.1. $\square$


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