The Stacks project

Lemma 73.9.2. Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is an fpqc covering.

Proof. We will show that an fppf covering is an fpqc covering, and then the rest follows from Lemma 73.7.2. Let $\{ f_ i : U_ i \to U\} _{i \in I}$ be an fppf covering of algebraic spaces over $S$. By definition this means that the $f_ i$ are flat which checks the first condition of Definition 73.9.1. To check the second, let $V \to U$ be a morphism with $V$ affine. We may choose an étale covering $\{ V_{ij} \to V \times _ U U_ i\} $ with $V_{ij}$ affine. Then the compositions $f_{ij} : V_{ij} \to V \times _ U U_ i \to V$ are flat and locally of finite presentation as compositions of such (Morphisms of Spaces, Lemmas 67.28.2, 67.30.3, 67.39.7, and 67.39.8). Hence these morphisms are open (Morphisms of Spaces, Lemma 67.30.6) and we see that $|V| = \bigcup _{i \in I} \bigcup _{j \in J_ i} f_{ij}(|V_{ij}|)$ is an open covering of $|V|$. Since $|V|$ is quasi-compact, this covering has a finite refinement. Say $V_{i_1j_1}, \ldots , V_{i_ Nj_ N}$ do the job. Then $\{ V_{i_ kj_ k} \to V\} _{k = 1, \ldots , N}$ is a standard fpqc covering of $V$ refinining the family $\{ U_ i \times _ U V \to V\} $. This finishes the proof. $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 73.9: Fpqc topology

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Your email address will not be published.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