The Stacks project

Lemma 65.6.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There exists a surjective étale morphism $U \to X$ where $U$ is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of $S$.

Proof. Let $V \to X$ be a surjective étale morphism. Let $V = \bigcup _{i \in I} V_ i$ be a Zariski open covering such that each $V_ i$ maps into an affine open of $S$. Then set $U = \coprod _{i \in I} V_ i$ with induced morphism $U \to V \to X$. This is étale and surjective as a composition of étale and surjective representable transformations of functors (via the general principle Spaces, Lemma 64.5.4 and Morphisms, Lemmas 29.9.2 and 29.36.3). $\square$

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Your email address will not be published.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 03FX. Beware of the difference between the letter 'O' and the digit '0'.