The Stacks project

Remark 35.32.8. At this point we have three possible definitions of what it means for a property $\mathcal{P}$ of morphisms to be “étale local on the source and target”:

  1. $\mathcal{P}$ is étale local on the source and $\mathcal{P}$ is étale local on the target,

  2. (the definition in the paper [Page 100, DM] by Deligne and Mumford) for every diagram

    \[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

    with surjective étale vertical arrows we have $\mathcal{P}(h) \Leftrightarrow \mathcal{P}(f)$, and

  3. $\mathcal{P}$ is étale local on the source-and-target.

In this section we have seen that (SP) $\Rightarrow $ (DM) $\Rightarrow $ (ST). The Examples 35.32.1 and 35.32.2 show that neither implication can be reversed. Finally, Lemma 35.32.6 shows that the difference disappears when looking at properties of morphisms which are stable under postcomposing with open immersions, which in practice will always be the case.

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Your email address will not be published.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 04R4. Beware of the difference between the letter 'O' and the digit '0'.