The Stacks project

Proof. Let $y \in Y$ be a point. By More on Morphisms, Lemma 37.17.1 there exists an integer $0 \geq m$ and a commutative diagram

where $U \subset X$ is open, $V = Y \cap U$, $\pi$ is étale, $V = \pi ^{-1}(\mathbf{A}^ m_ T)$, and $y \in V$. Denote $z \in \mathbf{A}^ m_ T$ the image of $y$. Then we have

by Lemma 50.2.2. Denote $x_1, \ldots , x_{m + 1}$ the coordinate functions on $\mathbf{A}^{m + 1}_ T$. Since the conditions (1) and (2) in Definition 50.15.1 do not depend on the choice of the local coordinate, it suffices to check the conditions (1) and (2) when $f$ is the image of $x_{m + 1}$ by the flat local ring homomorphism $\mathcal{O}_{\mathbf{A}^{m + 1}_ T, z} \to \mathcal{O}_{X, x}$. In this way we see that it suffices to check conditions (1) and (2) for $\mathbf{A}^ m_ T \subset \mathbf{A}^{m + 1}_ T$ and the point $z$. To prove this case we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $T = \mathop{\mathrm{Spec}}(B)$ are affine. Let $A \to B$ be the ring map corresponding to the morphism $T \to S$ and set $P = B[x_1, \ldots , x_{m + 1}]$ so that $\mathbf{A}^{m + 1}_ T = \mathop{\mathrm{Spec}}(P)$. We have

Hence the map $P \to \Omega _{P/A}$, $g \mapsto g \text{d}x_{m + 1}$ is a split injection and $x_{m + 1}$ is a nonzerodivisor on $\Omega ^ p_{P/A}$ for all $p \geq 0$. Localizing at the prime ideal corresponding to $z$ finishes the proof. $\square$