The Stacks project

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 37.23.1 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

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

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. $\square$