# The Stacks Project

## Tag: 025C

This tag has label etale-theorem-geometric-structure and it points to

The corresponding content:

Theorem 37.12.3. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. If $\varphi$ is étale at $x$, then there exist exist affine opens $V \subset Y$ and $U \subset X$ 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 25.37.14 allthough the statements differ slightly. $\square$

\begin{theorem}
\label{theorem-geometric-structure}
Let $\varphi : X \to Y$ be a morphism of schemes.
Let $x \in X$.
If $\varphi$ is \'etale at $x$, then there exist exist affine opens
$V \subset Y$ and $U \subset X$ 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}
allthough the statements differ slightly.
\end{proof}


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