Lemma 65.6.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a Zariski covering $X = \bigcup X_ i$ such that each algebraic space $X_ i$ has a surjective étale covering by an affine scheme. We may in addition assume each $X_ i$ maps into an affine open of $S$.

**Proof.**
By Lemma 65.6.1 we can find a surjective étale morphism $U = \coprod U_ i \to X$, with $U_ i$ affine and mapping into an affine open of $S$. Let $X_ i \subset X$ be the open subspace of $X$ such that $U_ i \to X$ factors through an étale surjective morphism $U_ i \to X_ i$, see Lemma 65.4.10. Since $U = \bigcup U_ i$ we see that $X = \bigcup X_ i$. As $U_ i \to X_ i$ is surjective it follows that $X_ i \to S$ maps into an affine open of $S$.
$\square$

## 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Your email address will not be published.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)