## Tag `04RH`

Chapter 58: Morphisms of Algebraic Spaces > Section 58.38: Étale morphisms

Definition 58.38.1. 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 7560–7567 (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}
```

## Comments (2)

## Add a comment on tag `04RH`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.