The Stacks project

Lemma 37.37.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Let $h \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$. Assume

  1. $f$ is smooth at $x$, and

  2. the image $\text{d}\overline{h}$ of $\text{d}h$ in

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

    is nonzero.

Then there exists an affine open neighbourhood $U \subset X$ of $x$ such that $h$ comes from $h \in \Gamma (U, \mathcal{O}_ U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \in D$ and $D \to S$ smooth.

Proof. As $f$ is smooth at $x$ we may assume, after replacing $X$ by an open neighbourhood of $x$ that $f$ is smooth. In particular we see that $f$ is flat and locally of finite presentation. By Lemma 37.22.1 we already know there exists an open neighbourhood $U \subset X$ of $x$ such that $h$ comes from $h \in \Gamma (U, \mathcal{O}_ U)$ and such that $D = V(h)$ is an effective Cartier divisor in $U$ with $x \in D$ and $D \to S$ flat and of finite presentation. By Morphisms, Lemma 29.32.15 we have a short exact sequence

\[ \mathcal{C}_{D/U} \to i^*\Omega _{U/S} \to \Omega _{D/S} \to 0 \]

where $i : D \to U$ is the closed immersion and $\mathcal{C}_{D/U}$ is the conormal sheaf of $D$ in $U$. As $D$ is an effective Cartier divisor cut out by $h \in \Gamma (U, \mathcal{O}_ U)$ we see that $\mathcal{C}_{D/U} = h \cdot \mathcal{O}_ S$. Since $U \to S$ is smooth the sheaf $\Omega _{U/S}$ is finite locally free, hence its pullback $i^*\Omega _{U/S}$ is finite locally free also. The first arrow of the sequence maps the free generator $h$ to the section $\text{d}h|_ D$ of $i^*\Omega _{U/S}$ which has nonzero value in the fibre $\Omega _{U/S, x} \otimes \kappa (x)$ by assumption. By right exactness of $\otimes \kappa (x)$ we conclude that

\[ \dim _{\kappa (x)} \left( \Omega _{D/S, x} \otimes \kappa (x) \right) = \dim _{\kappa (x)} \left( \Omega _{U/S, x} \otimes \kappa (x) \right) - 1. \]

By Morphisms, Lemma 29.34.14 we see that $\Omega _{U/S, x} \otimes \kappa (x)$ can be generated by at most $\dim _ x(U_ s)$ elements. By the displayed formula we see that $\Omega _{D/S, x} \otimes \kappa (x)$ can be generated by at most $\dim _ x(U_ s) - 1$ elements. Note that $\dim _ x(D_ s) = \dim _ x(U_ s) - 1$ for example because $\dim (\mathcal{O}_{D_ s, x}) = \dim (\mathcal{O}_{U_ s, x}) - 1$ by Algebra, Lemma 10.60.13 (also $D_ s \subset U_ s$ is effective Cartier, see Divisors, Lemma 31.18.1) and then using Morphisms, Lemma 29.28.1. Thus we conclude that $\Omega _{D/S, x} \otimes \kappa (x)$ can be generated by at most $\dim _ x(D_ s)$ elements and we conclude that $D \to S$ is smooth at $x$ by Morphisms, Lemma 29.34.14 again. After shrinking $U$ we get that $D \to S$ is smooth and we win. $\square$


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