Lemma 37.34.7. Let $S$ be a scheme. Let $\mathcal{U} = \{ S_ i \to S\} _{i \in I}$ be a smooth covering of $S$, see Topologies, Definition 34.5.1. Then there exists an étale covering $\mathcal{V} = \{ T_ j \to S\} _{j \in J}$ (see Topologies, Definition 34.4.1) which refines (see Sites, Definition 7.8.1) $\mathcal{U}$.

**Proof.**
For every $s \in S$ there exists an $i \in I$ such that $s$ is in the image of $S_ i \to S$. By Lemma 37.34.6 we can find an étale morphism $g_ s : T_ s \to S$ such that $s \in g_ s(T_ s)$ and such that $g_ s$ factors through $S_ i \to S$. Hence $\{ T_ s \to S\} $ is an étale covering of $S$ that refines $\mathcal{U}$.
$\square$

## Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #2539 by Anonymous on