The Stacks project

Remark 35.32.7. Using Lemma 35.32.6 and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are étale local on the source-and-target. In each case we list the lemma which implies the property is étale local on the source and the lemma which implies the property is étale local on the target. In each case the third assumption of Lemma 35.32.6 is trivial to check, and we omit it. Here is the list:

  1. flat, see Lemmas 35.27.1 and 35.23.15,

  2. locally of finite presentation, see Lemmas 35.28.1 and 35.23.11,

  3. locally finite type, see Lemmas 35.28.2 and 35.23.10,

  4. universally open, see Lemmas 35.28.4 and 35.23.4,

  5. syntomic, see Lemmas 35.29.1 and 35.23.26,

  6. smooth, see Lemmas 35.30.1 and 35.23.27,

  7. étale, see Lemmas 35.31.1 and 35.23.29,

  8. locally quasi-finite, see Lemmas 35.31.2 and 35.23.24,

  9. unramified, see Lemmas 35.31.3 and 35.23.28,

  10. G-unramified, see Lemmas 35.31.3 and 35.23.28, and

  11. add more here as needed.

Comments (1)

Comment #463 by Kestutis Cesnavicius on

'Surjective' could also be added here.

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