Unramified morphism

A morphism $f:Y\to X$ of schemes is unramified if it is locally of finite type and its sheaf of relative Kähler differentials vanishes:

Equivalently, for a locally finite type morphism, the diagonal

is an open immersion.

Affine-locally, $\operatorname{Spec}B\to\operatorname{Spec}A$ is unramified when $B$ is a finitely generated $A$-algebra and $\Omega_{B/A}=0$. This condition is local on source and target.

For a finite field extension $K/F$, the morphism $\operatorname{Spec}K\to\operatorname{Spec}F$ is unramified exactly when $K/F$ is separable.