The Stacks project

Remark 74.21.5. Using Lemma 74.21.4 and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is etale local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma 74.21.4 is trivial to check, and we omit it. Here is the list:

  1. étale, see Lemmas 74.19.1 and 74.11.28,

  2. locally quasi-finite, see Lemmas 74.19.2 and 74.11.24,

  3. unramified, see Lemmas 74.19.3 and 74.11.27, and

  4. add more here as needed.

Of course any property listed in Remark 74.20.5 is a fortiori an example that could be listed here.

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