The Stacks project

Lemma 73.8.2. Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is a ph covering.

Proof. We will show that an fppf covering is a ph covering, and then the rest follows from Lemma 73.7.2. Let $\{ X_ i \to X\} _{i \in I}$ be an fppf covering of algebraic spaces over a base scheme $S$. Let $U$ be an affine scheme and let $U \to X$ be a morphism. We can refine the fppf covering $\{ X_ i \times _ U U \to U\} _{i \in I}$ by an fppf covering $\{ T_ i \to U\} _{i \in I}$ where $T_ i$ is a scheme (Lemma 73.7.4). Then we can find a standard ph covering $\{ U_ j \to U\} _{j = 1, \ldots , m}$ refining $\{ T_ i \to U\} _{i \in I}$ by More on Morphisms, Lemma 37.48.7 (and the definition of ph coverings for schemes). Thus $\{ X_ i \to X\} _{i \in I}$ is a ph covering by definition. $\square$


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Your email address will not be published.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