Lemma 59.104.8. Let $S$ be a scheme. Then the category fibred in groupoids

whose fibre category over $U$ is the category $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ of sheaves on the small étale site of $U$ is a stack in groupoids.

Lemma 59.104.8. Let $S$ be a scheme. Then the category fibred in groupoids

\[ p : \mathcal{S} \longrightarrow (\mathit{Sch}/S)_ h \]

whose fibre category over $U$ is the category $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ of sheaves on the small étale site of $U$ is a stack in groupoids.

**Proof.**
To prove the lemma we will check conditions (1), (2), and (3) of More on Flatness, Lemma 38.37.13.

Condition (1) holds because we have glueing for sheaves (and Zariski coverings are étale coverings). See Sites, Lemma 7.26.4.

To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Then we have descent for $\{ X \to Y\} $ by either Lemma 59.104.5 or Lemma 59.104.3.

Condition (3) follows immediately from the more general Lemma 59.104.7. $\square$

Your email address will not be published.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 (0)