Theorem 41.12.3 (025C)—The Stacks project

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Theorem 41.12.3. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$ . Let $V \subset Y$ be an affine open neighbourhood of $\varphi (x)$ . If $\varphi$ is étale at $x$ , then there exist exists an affine open $U \subset X$ with $x \in U$ and $\varphi (U) \subset V$ such that we have the following diagram

$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \mathop{\mathrm{Spec}}(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \mathop{\mathrm{Spec}}(R) }$

where $j$ is an open immersion, and $f \in R[t]$ is monic.

Proof. This is equivalent to Morphisms, Lemma 29.36.14 although the statements differ slightly. See also, Varieties, Lemma 33.18.3 for a variant for unramified morphisms. $\square$

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