The Stacks project

Lemma 61.12.2. Let $T$ be a scheme. Let $\{ f_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent

  1. $\{ f_ i : T_ i \to T\} _{i \in I}$ is a pro-étale covering,

  2. each $f_ i$ is weakly étale and $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering,

  3. each $f_ i$ is weakly étale and for every affine open $U \subset T$ there exist quasi-compact opens $U_ i \subset T_ i$ which are almost all empty, such that $U = \bigcup f_ i(U_ i)$,

  4. each $f_ i$ is weakly étale and there exists an affine open covering $T = \bigcup _{\alpha \in A} U_\alpha $ and for each $\alpha \in A$ there exist $i_{\alpha , 1}, \ldots , i_{\alpha , n(\alpha )} \in I$ and quasi-compact opens $U_{\alpha , j} \subset T_{i_{\alpha , j}}$ such that $U_\alpha = \bigcup _{j = 1, \ldots , n(\alpha )} f_{i_{\alpha , j}}(U_{\alpha , j})$.

If $T$ is quasi-separated, these are also equivalent to

  1. each $f_ i$ is weakly étale, and for every $t \in T$ there exist $i_1, \ldots , i_ n \in I$ and quasi-compact opens $U_ j \subset T_{i_ j}$ such that $\bigcup _{j = 1, \ldots , n} f_{i_ j}(U_ j)$ is a (not necessarily open) neighbourhood of $t$ in $T$.

Proof. The equivalence of (1) and (2) is immediate from the definitions. Hence the lemma follows from Topologies, Lemma 34.9.2. $\square$

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Your email address will not be published.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