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Tag 02GL

Chapter 28: Morphisms of Schemes > Section 28.34: Étale morphisms

Description of the étale schemes over fields and fibres of étale morphisms.

Lemma 28.34.7. Fibres of étale morphisms.

  1. Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\rm Spec}(k)$ is étale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.
  2. If $f : X \to S$ is an étale morphism, then for every $s \in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable field extensions of $\kappa(s)$.

Proof. You can deduce this from Lemma 28.33.11 via Lemma 28.34.5 above. Here is a direct proof.

We will use Algebra, Lemma 10.141.4. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\rm Spec}(k)$ is étale. Conversely, suppose that $X \to \mathop{\rm Spec}(k)$ is étale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma 28.19.2 for example). Thus $X$ is a discrete space and we win. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 7508–7524 (see updates for more information).

    \begin{lemma}
    \label{lemma-etale-over-field}
    \begin{slogan}
    Description of the \'etale schemes over fields and fibres
    of \'etale morphisms.
    \end{slogan}
    Fibres of \'etale morphisms.
    \begin{enumerate}
    \item Let $X$ be a scheme over a field $k$.
    The structure morphism $X \to \Spec(k)$ is \'etale if
    and only if $X$ is a disjoint union of spectra of finite separable
    field extensions of $k$.
    \item If $f : X \to S$ is an \'etale morphism, then for every $s \in S$ the
    fibre $X_s$ is a disjoint union of spectra of finite separable field
    extensions of $\kappa(s)$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    You can deduce this from Lemma \ref{lemma-unramified-over-field}
    via Lemma \ref{lemma-etale-smooth-unramified} above.
    Here is a direct proof.
    
    \medskip\noindent
    We will use Algebra, Lemma \ref{algebra-lemma-etale-over-field}.
    Hence it is clear that if $X$ is a disjoint union of spectra of finite
    separable field extensions of $k$ then $X \to \Spec(k)$ is \'etale.
    Conversely, suppose that $X \to \Spec(k)$ is \'etale. Then for any affine
    open $U \subset X$ we see that $U$ is a finite disjoint union of spectra
    of finite separable field extensions of $k$. Hence all points of $X$
    are closed points (see
    Lemma \ref{lemma-algebraic-residue-field-extension-closed-point-fibre}
    for example). Thus $X$ is a discrete space and we win.
    \end{proof}

    Comments (1)

    Comment #1039 by Jakob Scholbach on September 16, 2014 a 1:04 pm UTC

    Suggested slogan: Description of the ├ętale site of a field.

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