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Tag 025C

Chapter 40: Étale Morphisms of Schemes > Section 40.12: The structure theorem

Theorem 40.12.3. 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 28.37.14 although the statements differ slightly. $\square$

    The code snippet corresponding to this tag is a part of the file etale.tex and is located in lines 1239–1253 (see updates for more information).

    \begin{theorem}
    \label{theorem-geometric-structure}
    Let $\varphi : X \to Y$ be a morphism of schemes.
    Let $x \in X$.
    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}
    although the statements differ slightly.
    \end{proof}

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