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

where $\pi $ is étale.

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