According to J. S. Milne, Étale cohomology:
A morphism $f: Y \longrightarrow X$ that is locally of finite-type is said to be unramified at $y \in Y$ if $\mathcal{O}_{Y,y}/\mathfrak{m}_{x}\mathcal{O}_{Y,y}$ is finite separable field extension of $k(x)$, where $x=f(y)$.
My question is this: Based on the definition above, if I have a finite morphism $f: Y \longrightarrow X$ of projective varieties, I could say directly that:
"A finite morphism $f: Y \longrightarrow X$ of projective varieties is said to be unramified at $y \in Y$ if $\mathcal{O}_{Y,y}/\mathfrak{m}_{x}\mathcal{O}_{Y,y}$ is finite separable field extension of $k(x)$, where $x=f(y)$" Or I need to add hypotheses about Y and X, be like normal varieties, for example?