The Stacks project

Remark 65.7.3. Here is a list of properties which are local for the ├ętale topology (keep in mind that the fpqc, fppf, syntomic, and smooth topologies are stronger than the ├ętale 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. Gorenstein, see Duality for Schemes, Lemma 48.24.6,

  6. reduced, see Descent, Lemma 35.18.1,

  7. normal, see Descent, Lemma 35.18.2,

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

  9. regular, see Descent, Lemma 35.18.4,

  10. Nagata, see Descent, Lemma 35.18.5.

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