The Stacks project

41.13 Étale and smooth morphisms

An étale morphism is smooth of relative dimension zero. The projection $\mathbf{A}^ n_ S \to S$ is a standard example of a smooth morphism of relative dimension $n$. It turns out that any smooth morphism is étale locally of this form. Here is the precise statement.

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. $\square$

Comments (2)

Comment #3240 by Dario Weißmann on

Typo in the statement: then there exist and integer

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