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Tag 03XT

Chapter 55: Morphisms of Algebraic Spaces > Section 55.38: Étale morphisms

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. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 7556–7595 (see updates for more information).

    \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}

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