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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