## Tag `03FR`

Chapter 57: Properties of Algebraic Spaces > Section 57.16: Étale morphisms of algebraic spaces

Definition 57.16.2. Let $S$ be a scheme. A morphism $f : X \to Y$ between algebraic spaces over $S$ is called

étaleif and only if for every étale morphism $\varphi : U \to X$ where $U$ is a scheme, the composition $f \circ \varphi$ is étale also.

The code snippet corresponding to this tag is a part of the file `spaces-properties.tex` and is located in lines 2083–2090 (see updates for more information).

```
\begin{definition}
\label{definition-etale}
Let $S$ be a scheme.
A morphism $f : X \to Y$ between algebraic spaces over $S$ is
called {\it \'etale} if and only if for every \'etale morphism
$\varphi : U \to X$ where $U$ is a scheme, the composition
$f \circ \varphi$ is \'etale also.
\end{definition}
```

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