The Stacks project

Lemma 61.12.4. Let $T$ be a scheme.

  1. If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a pro-étale covering of $T$.

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

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

Proof. This follows from the fact that composition and base changes of weakly étale morphisms are weakly étale (More on Morphisms, Lemmas 37.62.5 and 37.62.6), Lemma 61.12.2, and the corresponding results for fpqc coverings, see Topologies, Lemma 34.9.7. $\square$

Comments (0)

There are also:

  • 6 comment(s) on Section 61.12: The pro-étale site

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Your email address will not be published.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 098C. Beware of the difference between the letter 'O' and the digit '0'.