The Stacks project

Lemma 37.37.2. 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 smooth at $x$, and

  2. the map

    \[ \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) \longrightarrow \Omega _{\kappa (x)/\kappa (s)} \]

    has a nonzero kernel.

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 smooth.

Proof. Write $k = \kappa (s)$ and $R = \mathcal{O}_{X_ s, x}$. Denote $\mathfrak m$ the maximal ideal of $R$ and $\kappa = R/\mathfrak m$ so that $\kappa = \kappa (x)$. As formation of modules of differentials commutes with localization (see Algebra, Lemma 10.131.8) we have $\Omega _{X_ s/s, x} = \Omega _{R/k}$. By Algebra, Lemma 10.131.9 there is an exact sequence

\[ \mathfrak m/\mathfrak m^2 \xrightarrow {\text{d}} \Omega _{R/k} \otimes _ R \kappa \to \Omega _{\kappa /k} \to 0. \]

Hence if (2) holds, there exists an element $\overline{h} \in \mathfrak m$ such that $\text{d}\overline{h}$ is nonzero. Choose a lift $h \in \mathcal{O}_{X, x}$ of $\overline{h}$ and apply Lemma 37.37.1. $\square$


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