Remark 100.7.3. Here is a list of properties which are local for the smooth topology (keep in mind that the fpqc, fppf, and syntomic topologies are stronger than the smooth topology):

locally Noetherian, see Descent, Lemma 35.16.1,

Jacobson, see Descent, Lemma 35.16.2,

locally Noetherian and $(S_ k)$, see Descent, Lemma 35.17.1,

Cohen-Macaulay, see Descent, Lemma 35.17.2,

reduced, see Descent, Lemma 35.18.1,

normal, see Descent, Lemma 35.18.2,

locally Noetherian and $(R_ k)$, see Descent, Lemma 35.18.3,

regular, see Descent, Lemma 35.18.4,

Nagata, see Descent, Lemma 35.18.5.

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