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.