The Stacks project

Remark 68.4.1. Before we give the proof of the next lemma let us recall some facts about étale morphisms of schemes:

  1. An étale morphism is flat and hence generalizations lift along an étale morphism (Morphisms, Lemmas 29.36.12 and 29.25.9).

  2. An étale morphism is unramified, an unramified morphism is locally quasi-finite, hence fibres are discrete (Morphisms, Lemmas 29.36.16, 29.35.10, and 29.20.6).

  3. A quasi-compact étale morphism is quasi-finite and in particular has finite fibres (Morphisms, Lemmas 29.20.9 and 29.20.10).

  4. An étale scheme over a field $k$ is a disjoint union of spectra of finite separable field extension of $k$ (Morphisms, Lemma 29.36.7).

For a general discussion of étale morphisms, please see Étale Morphisms, Section 41.11.

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