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Tag 03XS

55.38. Étale morphisms

The notion of an étale morphism of algebraic spaces was defined in Properties of Spaces, Definition 54.15.2. Here is what it means for a morphism to be étale at a point.

Definition 55.38.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. We say $f$ is étale at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is étale

Lemma 55.38.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is étale,
  2. for every $x \in |X|$ the morphism $f$ is étale at $x$,
  3. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is étale,
  4. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is étale,
  5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times_Y X \to V$ is an étale morphism,
  6. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is étale,
  7. for every commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is étale,
  8. there exists a commutative diagram $$ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ surjective such that the top horizontal arrow is étale, and
  9. there exist Zariski coverings $Y = \bigcup Y_i$ and $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_i$ is étale.

Proof. Combine Properties of Spaces, Lemmas 54.15.3, 54.15.5 and 54.15.4. Some details omitted. $\square$

Lemma 55.38.3. The composition of two étale morphisms of algebraic spaces is étale.

Proof. This is a copy of Properties of Spaces, Lemma 54.15.4. $\square$

Lemma 55.38.4. The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.

Proof. This is a copy of Properties of Spaces, Lemma 54.15.5. $\square$

Lemma 55.38.5. An étale morphism of algebraic spaces is locally quasi-finite.

Proof. Let $X \to Y$ be an étale morphism of algebraic spaces, see Properties of Spaces, Definition 54.15.2. By Properties of Spaces, Lemma 54.15.3 we see this means there exists a diagram as in Lemma 55.22.1 with $h$ étale and surjective vertical arrow $a$. By Morphisms, Lemma 28.34.6 $h$ is locally quasi-finite. Hence $X \to Y$ is locally quasi-finite by definition. $\square$

Lemma 55.38.6. An étale morphism of algebraic spaces is smooth.

Proof. The proof is identical to the proof of Lemma 55.38.5. It uses the fact that an étale morphism of schemes is smooth (by definition of an étale morphism of schemes). $\square$

Lemma 55.38.7. An étale morphism of algebraic spaces is flat.

Proof. The proof is identical to the proof of Lemma 55.38.5. It uses Morphisms, Lemma 28.34.12. $\square$

Lemma 55.38.8. An étale morphism of algebraic spaces is locally of finite presentation.

Proof. The proof is identical to the proof of Lemma 55.38.5. It uses Morphisms, Lemma 28.34.11. $\square$

Lemma 55.38.9. An étale morphism of algebraic spaces is locally of finite type.

Proof. An étale morphism is locally of finite presentation and a morphism locally of finite presentation is locally of finite type, see Lemmas 55.38.8 and 55.28.5. $\square$

Lemma 55.38.10. An étale morphism of algebraic spaces is unramified.

Proof. The proof is identical to the proof of Lemma 55.38.5. It uses Morphisms, Lemma 28.34.5. $\square$

Lemma 55.38.11. Let $S$ be a scheme. Let $X, Y$ be algebraic spaces étale over an algebraic space $Z$. Any morphism $X \to Y$ over $Z$ is étale.

Proof. This is a copy of Properties of Spaces, Lemma 54.15.6. $\square$

Lemma 55.38.12. A locally finitely presented, flat, unramified morphism of algebraic spaces is étale.

Proof. Let $X \to Y$ be a locally finitely presented, flat, unramified morphism of algebraic spaces. By Properties of Spaces, Lemma 54.15.3 we see this means there exists a diagram as in Lemma 55.22.1 with $h$ locally finitely presented, flat, unramified and surjective vertical arrow $a$. By Morphisms, Lemma 28.34.16 $h$ is étale. Hence $X \to Y$ is étale by definition. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 7539–7743 (see updates for more information).

    \section{\'Etale morphisms}
    \label{section-etale}
    
    \noindent
    The notion of an \'etale morphism of algebraic spaces was defined in
    Properties of Spaces, Definition \ref{spaces-properties-definition-etale}.
    Here is what it means for a morphism to be \'etale at a point.
    
    \begin{definition}
    \label{definition-etale}
    Let $S$ be a scheme.
    Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
    Let $x \in |X|$. We say $f$ is {\it \'etale at $x$} if there
    exists an open neighbourhood $X' \subset X$ of $x$ such that
    $f|_{X'} : X' \to Y$ is \'etale
    \end{definition}
    
    \begin{lemma}
    \label{lemma-etale-local}
    Let $S$ be a scheme.
    Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
    The following are equivalent:
    \begin{enumerate}
    \item $f$ is \'etale,
    \item for every $x \in |X|$ the morphism $f$ is \'etale at $x$,
    \item for every scheme $Z$ and any morphism $Z \to Y$ the morphism
    $Z \times_Y X \to Z$ is \'etale,
    \item for every affine scheme $Z$ and any morphism
    $Z \to Y$ the morphism $Z \times_Y X \to Z$ is \'etale,
    \item there exists a scheme $V$ and a surjective \'etale morphism
    $V \to Y$ such that $V \times_Y X \to V$ is an \'etale morphism,
    \item there exists a scheme $U$ and a surjective \'etale morphism
    $\varphi : U \to X$ such that the composition $f \circ \varphi$
    is \'etale,
    \item for every commutative diagram
    $$
    \xymatrix{
    U \ar[d] \ar[r] & V \ar[d] \\
    X \ar[r] & Y
    }
    $$
    where $U$, $V$ are schemes and the vertical arrows are \'etale
    the top horizontal arrow is \'etale,
    \item there exists a commutative diagram
    $$
    \xymatrix{
    U \ar[d] \ar[r] & V \ar[d] \\
    X \ar[r] & Y
    }
    $$
    where $U$, $V$ are schemes, the vertical arrows are \'etale, and
    $U \to X$ surjective such that the top horizontal arrow is \'etale, and
    \item there exist Zariski coverings $Y = \bigcup Y_i$ and
    $f^{-1}(Y_i) = \bigcup X_{ij}$ such that each morphism
    $X_{ij} \to Y_i$ is \'etale.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Combine
    Properties of Spaces, Lemmas
    \ref{spaces-properties-lemma-etale-local},
    \ref{spaces-properties-lemma-base-change-etale} and
    \ref{spaces-properties-lemma-composition-etale}.
    Some details omitted.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-composition-etale}
    The composition of two \'etale morphisms of algebraic spaces
    is \'etale.
    \end{lemma}
    
    \begin{proof}
    This is a copy of
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-composition-etale}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-base-change-etale}
    The base change of an \'etale morphism of algebraic spaces
    by any morphism of algebraic spaces is \'etale.
    \end{lemma}
    
    \begin{proof}
    This is a copy of
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-base-change-etale}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-locally-quasi-finite}
    An \'etale morphism of algebraic spaces is locally quasi-finite.
    \end{lemma}
    
    \begin{proof}
    Let $X \to Y$ be an \'etale morphism of algebraic spaces, see
    Properties of Spaces, Definition \ref{spaces-properties-definition-etale}.
    By
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-local}
    we see this means there exists a diagram as in
    Lemma \ref{lemma-local-source-target}
    with $h$ \'etale and surjective vertical arrow $a$. By
    Morphisms, Lemma \ref{morphisms-lemma-etale-locally-quasi-finite}
    $h$ is locally quasi-finite. Hence $X \to Y$ is locally quasi-finite
    by definition.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-smooth}
    An \'etale morphism of algebraic spaces is smooth.
    \end{lemma}
    
    \begin{proof}
    The proof is identical to the proof of
    Lemma \ref{lemma-etale-locally-quasi-finite}.
    It uses the fact that an \'etale morphism of schemes is smooth
    (by definition of an \'etale morphism of schemes).
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-flat}
    An \'etale morphism of algebraic spaces is flat.
    \end{lemma}
    
    \begin{proof}
    The proof is identical to the proof of
    Lemma \ref{lemma-etale-locally-quasi-finite}.
    It uses
    Morphisms, Lemma \ref{morphisms-lemma-etale-flat}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-locally-finite-presentation}
    \begin{slogan}
    \'Etale implies locally of finite presentation.
    \end{slogan}
    An \'etale morphism of algebraic spaces is locally of finite presentation.
    \end{lemma}
    
    \begin{proof}
    The proof is identical to the proof of
    Lemma \ref{lemma-etale-locally-quasi-finite}.
    It uses
    Morphisms, Lemma \ref{morphisms-lemma-etale-locally-finite-presentation}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-locally-finite-type}
    An \'etale morphism of algebraic spaces is locally of finite type.
    \end{lemma}
    
    \begin{proof}
    An \'etale morphism is locally of finite presentation
    and a morphism locally of finite presentation is locally of finite type,
    see
    Lemmas \ref{lemma-etale-locally-finite-presentation} and
    \ref{lemma-finite-presentation-finite-type}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-unramified}
    An \'etale morphism of algebraic spaces is unramified.
    \end{lemma}
    
    \begin{proof}
    The proof is identical to the proof of
    Lemma \ref{lemma-etale-locally-quasi-finite}.
    It uses
    Morphisms, Lemma \ref{morphisms-lemma-etale-smooth-unramified}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-etale-permanence}
    Let $S$ be a scheme. Let $X, Y$ be algebraic spaces \'etale over
    an algebraic space $Z$. Any morphism $X \to Y$ over $Z$ is \'etale.
    \end{lemma}
    
    \begin{proof}
    This is a copy of
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-permanence}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-unramified-flat-lfp-etale}
    A locally finitely presented, flat, unramified morphism of algebraic
    spaces is \'etale.
    \end{lemma}
    
    \begin{proof}
    Let $X \to Y$ be a locally finitely presented, flat, unramified morphism
    of algebraic spaces. By
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-local}
    we see this means there exists a diagram as in
    Lemma \ref{lemma-local-source-target}
    with $h$ locally finitely presented, flat, unramified
    and surjective vertical arrow $a$. By
    Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}
    $h$ is \'etale. Hence $X \to Y$ is \'etale by definition.
    \end{proof}

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