Remark 40.6.3. Notwithstanding the warning in Remark 40.6.2 there are some cases where Lemma 40.6.1 can be used without causing too much 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:
$\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“relative dimension $\leq d$”. See Morphisms, Definition 29.29.1 and Morphisms, Lemmas 29.28.4 and 29.28.3.
$\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“locally quasi-finite”. This is the case $d = 0$ of the previous item, see Morphisms, Lemma 29.29.5.
$\mathcal{Q} = \mathcal{R} =$“locally of finite type”, and $\mathcal{P} =$“unramified”. See Morphisms, Lemmas 29.35.3 and 29.35.15.
What is interesting about the cases listed above is that we do not need to assume that $s, t$ are flat to get a conclusion about the locus where the morphism $h$ has property $\mathcal{P}$. We continue the list:
$\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P} =$“flat”. See More on Morphisms, Theorem 37.15.1 and Lemma 37.15.2.
$\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“Cohen-Macaulay”. See More on Morphisms, Definition 37.22.1 and More on Morphisms, Lemmas 37.22.6 and 37.22.7.
$\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“syntomic” use Morphisms, Lemma 29.30.12 (the locus is automatically open).
$\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“smooth”. See Morphisms, Lemma 29.34.15 (the locus is automatically open).
$\mathcal{Q} =$“locally of finite presentation”, $\mathcal{R} =$“flat and locally of finite presentation”, and $\mathcal{P}=$“étale”. See Morphisms, Lemma 29.36.17 (the locus is automatically open).
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