The Stacks project

Remark 72.19.5. Using Lemma 72.19.4 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 smooth 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 72.19.4 is trivial to check, and we omit it. Here is the list:

  1. flat, see Lemmas 72.14.1 and 72.10.13,

  2. locally of finite presentation, see Lemmas 72.15.1 and 72.10.10,

  3. locally finite type, see Lemmas 72.15.2 and 72.10.9,

  4. universally open, see Lemmas 72.15.4 and 72.10.4,

  5. syntomic, see Lemmas 72.16.1 and 72.10.25,

  6. smooth, see Lemmas 72.17.1 and 72.10.26,

  7. add more here as needed.


Comments (5)

Comment #462 by Kestutis Cesnavicius on

One can add 'surjective' to the list (although I couldn't find the (obvious) lemma that would tell that this is local on the source).

Comment #481 by on

Hi, why do you think that "surjective" is local on the source? If you look at Definition 72.13.1 it should be clear that it is not local on the source.

Comment #492 by Kestutis Cesnavicius on

Hi. Sorry, I haven't checked that definition and didn't realize that in saying "local on the source" you don't take a big disjoint union to have a covering of one element and then check that P holds for the composed morphism. With such a definition, which is adopted in Laumon and Moret-Bailly's book (see p. 33), 'surjective' would be local on the source. I wonder why they chose a different definition in "Champs algebriques"?

P.S. A possibility of automatically getting an email upon a follow up comment would be great---it's sometimes difficult to notice follow up comments.

Comment #495 by on

Well, (temporarily) subscribing to the RSS feed may be an alternative to having email notification.

Comment #496 by on

Just looked up the definition in Laumon and Moret-Bailly. Weird! In Brian Osserman's notes on the affine communication lemma and in Ulrich Görtz, Torsten Wedhorn book, they choose similarly to what is done in the Stacks project.

The reason for picking our definition of "P is local on the source" the way we do is that we want it to be true that you can check P "locally on the source", in other words, a morphism has property P if and only if every point of has an open neighbourghood such that has propery P.


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