The Stacks Project

Tag 03XT

Lemma 58.38.2. 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 57.15.3, 57.15.5 and 57.15.4. 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 7569–7608 (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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