Theorem 41.13.1 (039Q)—The Stacks project

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Theorem 41.13.1. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$ . If $\varphi$ is smooth at $x$ , then there exist an integer $n \geq 0$ and affine opens $V \subset Y$ and $U \subset X$ with $x \in U$ and $\varphi (U) \subset V$ such that there exists a commutative diagram

$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^ n_ R \ar[d] \ar@{=}[r] & \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]) \ar[dl] \\ Y & V \ar[l] \ar@{=}[r] & \mathop{\mathrm{Spec}}(R) }$

where $\pi$ is étale.

Proof. See Morphisms, Lemma 29.36.20. $\square$

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