Lemma 72.7.4. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{ f_ i : X_ i \to X\} _{i \in I}$ is an fppf covering of $X$. Then there exists a refinement $\mathcal{V} = \{ g_ i : T_ i \to X\} $ of $\mathcal{U}$ which is an fppf covering such that each $T_ i$ is a scheme.

**Proof.**
Omitted. Hint: For each $i$ choose a scheme $T_ i$ and a surjective étale morphism $T_ i \to X_ i$. Then check that $\{ T_ i \to X\} $ is an fppf covering.
$\square$

## 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Your email address will not be published.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)