Lemma 76.9.10. The property of being a thickening is fpqc local. Similarly for first order thickenings.

**Proof.**
The statement means the following: Let $S$ be a scheme and let $X \to X'$ be a morphism of algebraic spaces over $S$. Let $\{ g_ i : X'_ i \to X'\} $ be an fpqc covering of algebraic spaces such that the base change $X_ i \to X'_ i$ is a thickening for all $i$. Then $X \to X'$ is a thickening. Since the morphisms $g_ i$ are jointly surjective we conclude that $X \to X'$ is surjective. By Descent on Spaces, Lemma 74.11.17 we conclude that $X \to X'$ is a closed immersion. Thus $X \to X'$ is a thickening. We omit the proof in the case of first order thickenings.
$\square$

## 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Your email address will not be published.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 (0)