The Stacks project

Proposition 59.26.2. Facts on étale morphisms.

  1. Let $k$ be a field. A morphism of schemes $U \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if $U \cong \coprod _{i \in I} \mathop{\mathrm{Spec}}(k_ i)$ such that for each $i \in I$ the ring $k_ i$ is a field which is a finite separable extension of $k$.

  2. Let $\varphi : U \to S$ be a morphism of schemes. The following conditions are equivalent:

    1. $\varphi $ is étale,

    2. $\varphi $ is locally finitely presented, flat, and all its fibres are étale,

    3. $\varphi $ is flat, unramified and locally of finite presentation.

  3. A ring map $A \to B$ is étale if and only if $B \cong A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$ such that $\Delta = \det \left( \frac{\partial f_ i}{\partial x_ j} \right)$ is invertible in $B$.

  4. The base change of an étale morphism is étale.

  5. Compositions of étale morphisms are étale.

  6. Fibre products and products of étale morphisms are étale.

  7. An étale morphism has relative dimension 0.

  8. Let $Y \to X$ be an étale morphism. If $X$ is reduced (respectively regular) then so is $Y$.

  9. Étale morphisms are open.

  10. If $X \to S$ and $Y \to S$ are étale, then any $S$-morphism $X \to Y$ is also étale.

Proof. We have proved these facts (and more) in the preceding chapters. Here is a list of references: (1) Morphisms, Lemma 29.36.7. (2) Morphisms, Lemmas 29.36.8 and 29.36.16. (3) Algebra, Lemma 10.143.2. (4) Morphisms, Lemma 29.36.4. (5) Morphisms, Lemma 29.36.3. (6) Follows formally from (4) and (5). (7) Morphisms, Lemmas 29.36.6 and 29.29.5. (8) See Algebra, Lemmas 10.163.7 and 10.163.5, see also more results of this kind in Étale Morphisms, Section 41.19. (9) See Morphisms, Lemma 29.25.10 and 29.36.12. (10) See Morphisms, Lemma 29.36.18. $\square$

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