Étale morphism

A morphism $f:Y\to X$ of schemes is étale if it is simultaneously

• flat,
• unramified, and
• locally of finite presentation.

These conditions are local on both source and target, so étaleness may be checked on affine open covers. For an affine morphism $\operatorname{Spec}B\to\operatorname{Spec}A$, this says that $B$ is a flat, finitely presented $A$-algebra with $\Omega_{B/A}=0$.

For a finite field extension $K/F$, the morphism $\operatorname{Spec}K\to\operatorname{Spec}F$ is étale exactly when $K/F$ is separable. Étale morphisms are therefore the algebraic-geometric analogue of local diffeomorphisms.