The Stacks project

Lemma 38.34.11. Let $\mathit{Sch}_ h$ be a big h site as in Definition 38.34.9. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary h covering of $T$.

  1. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_ h$ which refines $\{ T_ i \to T\} _{i \in I}$.

  2. If $\{ T_ i \to T\} _{i \in I}$ is a standard h covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_ h$.

  3. If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_ h$.

Proof. Omitted. Hint: this is exactly the same as the proof of Topologies, Lemma 34.8.10. $\square$


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Your email address will not be published.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 0EU0. Beware of the difference between the letter 'O' and the digit '0'.