The Stacks project

Proof. By More on Morphisms, Lemma 37.22.7 we see that $Y \to S$ is a Cohen-Macaulay morphism. By Morphisms, Lemma 29.29.4 we see that $Y \to S$ has relative dimension $d$. By Divisors, Lemma 31.10.3 the $d$th Fitting ideal $\mathcal{I} \subset \mathcal{O}_ Y$ of $\Omega _{Y/S}$ cuts out the singular locus of the morphism $Y \to S$. In other words, $V(\mathcal{I}) \subset Y$ is the closed subset of points where $Y \to S$ is not smooth. By Divisors, Lemma 31.10.1 formation of this Fitting ideal commutes with base change. By assumption we see that $t^{n - 1}$ is a section of $\mathcal{I} + t^ n\mathcal{O}_ Y$. Thus for every $x \in X = V(t) \subset Y$ we conclude that $t^{n - 1} \in \mathcal{I}_ x$ where $\mathcal{I}_ x$ is the stalk at $x$. This implies that $V(\mathcal{I}) \subset V(t)$ in an open neighbourhood of $X$ in $Y$. Since $Y \to S$ is proper, this implies $V(\mathcal{I}) \subset V(t)$ as desired. $\square$