The Stacks project

Lemma 81.4.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\pi : X' \to X$ be a morphism of algebraic spaces. Assume

  1. $f \circ \pi $ is representable (by schemes),

  2. $f \circ \pi $ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.

Then

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

Proof. Formal consequence of Lemma 81.4.3 and Stacks, Lemma 8.3.7. $\square$

Comments (0)

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GFV. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 81.4.4 as pdf