The Stacks project

Definition 41.11.4. (See Morphisms, Definition 29.36.1.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type.

1. Let $x \in X$. We say $f$ is étale at $x \in X$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an étale homomorphism of local rings.

2. The morphism is said to be étale if it is étale at all its points.