The Stacks project

Lemma 76.9.6. Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. The functor

\[ V' \longmapsto V = X \times _{X'} V' \]

defines an equivalence of categories $X'_{\acute{e}tale}\to X_{\acute{e}tale}$.

Proof. The functor $V' \mapsto V$ defines an equivalence of categories $X'_{spaces, {\acute{e}tale}} \to X_{spaces, {\acute{e}tale}}$, see Theorem 76.8.1. Thus it suffices to show that $V$ is a scheme if and only if $V'$ is a scheme. This is the content of Lemma 76.9.5. $\square$

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Your email address will not be published.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 05ZS. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 76.9.6 as pdf