## 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:

- $f$ is étale,
- for every $x \in |X|$ the morphism $f$ is étale at $x$,
- for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is étale,
- for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times_Y X \to Z$ is étale,
- 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,
- there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is étale,
- 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,
- 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
- 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 7546–7585 (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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