The Stacks Project

Tag 025C

Theorem 40.12.3. 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 $$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \mathop{\rm Spec}(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \mathop{\rm Spec}(R) }$$ where $j$ is an open immersion, and $f \in R[t]$ is monic.

Proof. This is equivalent to Morphisms, Lemma 28.34.14 although the statements differ slightly. See also, Varieties, Lemma 32.18.3 for a variant for unramified morphisms. $\square$

The code snippet corresponding to this tag is a part of the file etale.tex and is located in lines 1299–1313 (see updates for more information).

\begin{theorem}
\label{theorem-geometric-structure}
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 \'etale 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
$$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \Spec(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \Spec(R) }$$
where $j$ is an open immersion, and $f \in R[t]$ is monic.
\end{theorem}

\begin{proof}
This is equivalent to
Morphisms, Lemma \ref{morphisms-lemma-etale-locally-standard-etale}
although the statements differ slightly.
See also, Varieties, Lemma \ref{varieties-lemma-geometric-structure-unramified}
for a variant for unramified morphisms.
\end{proof}

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