The Stacks project

Remark 99.9.20. Notwithstanding the warning in Remark 99.9.19 there are some cases where Lemma 99.9.18 can be used without causing ambiguity. We give a list. In each case we omit the verification of assumptions (1) and (2) and we give references which imply (3) and (4). Here is the list:

  1. $\mathcal{Q} = $“locally of finite type”, $\mathcal{R} = \emptyset $, and $\mathcal{P} =$“relative dimension $\leq d$”. See Morphisms of Spaces, Definition 66.33.2 and Morphisms of Spaces, Lemmas 66.34.4 and 66.34.3.

  2. $\mathcal{Q} =$“locally of finite type”, $\mathcal{R} = \emptyset $, and $\mathcal{P} =$“locally quasi-finite”. This is the case $d = 0$ of the previous item, see Morphisms of Spaces, Lemma 66.34.6. On the other hand, properties (3) and (4) are spelled out in Morphisms of Spaces, Lemma 66.34.7.

  3. $\mathcal{Q} = $“locally of finite type”, $\mathcal{R} = \emptyset $, and $\mathcal{P} =$“unramified”. This is Morphisms of Spaces, Lemma 66.38.10.

  4. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P} =$“flat”. See More on Morphisms of Spaces, Theorem 75.22.1 and Lemma 75.22.2. Note that here $W(\mathcal{P}, f)$ is always exactly the set of points where the morphism $f$ is flat because we only consider this open when $f$ has $\mathcal{Q}$ (see loc.cit.).

  5. $\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“étale”. This follows on combining (3) and (4) because an unramified morphism which is flat and locally of finite presentation is étale, see Morphisms of Spaces, Lemma 66.39.12.

  6. Add more here as needed (compare with the longer list at More on Groupoids, Remark 40.6.3).

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