Unramified morphism

Remarks

An open immersion or a closed immersion is unramified. For a closed immersion, each $\mathcal{O}_{X,x}$ is a quotient of $\mathcal{O}_{Y,y}$, so both conditions hold with $\kappa(x) = \kappa(y)$. For a field $k$ of characteristic not $2$, the map $\operatorname{Spec} k[t] \to \operatorname{Spec} k[s]$ given by $s \mapsto t^2$ is not unramified at $t = 0$. There $\mathfrak{m}_y \mathcal{O}_{X,x} = (t^2)$, which is strictly smaller than $\mathfrak{m}_x = (t)$, so condition 1 fails. For $f$ locally of finite type, $f$ is unramified exactly when the sheaf of relative Kähler differentials $\Omega_{X/Y}$ is zero. Some authors, as in EGA, also require $f$ to be locally of finite presentation. The ring version of this notion is an unramified morphism of rings.