## Tag `04RH`

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

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.

The code snippet corresponding to this tag is a part of the file `spaces-morphisms.tex` and is located in lines 7537–7544 (see updates for more information).

```
\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}
```

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