The Stacks project

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):

1. locally Noetherian, see Descent, Lemma 35.16.1,

2. Jacobson, see Descent, Lemma 35.16.2,

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

4. Cohen-Macaulay, see Descent, Lemma 35.17.2,

5. reduced, see Descent, Lemma 35.18.1,

6. normal, see Descent, Lemma 35.18.2,

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

8. regular, see Descent, Lemma 35.18.4,

9. Nagata, see Descent, Lemma 35.18.5.