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.
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