The Stacks project

Lemma 34.9.10. Let $T$ be an affine scheme.

  1. If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a standard fpqc covering of $T$.

  2. If $\{ T_ i \to T\} _{i\in I}$ is a standard fpqc covering and for each $i$ we have a standard fpqc covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard fpqc covering.

  3. If $\{ T_ i \to T\} _{i\in I}$ is a standard fpqc covering and $T' \to T$ is a morphism of affine schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a standard fpqc covering.

Proof. This follows formally from the fact that compositions and base changes of flat morphisms are flat (Morphisms, Lemmas 29.25.8 and 29.25.6) and that fibre products of affine schemes are affine (Schemes, Lemma 26.17.2). $\square$


Comments (0)

There are also:

  • 6 comment(s) on Section 34.9: The 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Your email address will not be published.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 03LA. Beware of the difference between the letter 'O' and the digit '0'.