The Stacks project

Remark 35.29.8. 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 35.29.1 and 35.29.2 show that neither implication can be reversed. Finally, Lemma 35.29.6 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.


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