According to Hartshorne's AG text, p. 300 Proposition 2.1, $\Omega_{X/Y} = 0$ at $\eta = $ generic point of $X$ follows from (II, 8.6A), which states that $$ \Omega_{K/k}=0 $$ if $K/k$ is separable.
Now, in Prop. 2.1 $K(X)/K(Y)$ is separable, but if I am not mistaken then $\Omega_{X/Y,\eta} = \Omega_{K(X)/A}$ for some appropriate open subset $U = \text{Spec}A$ of $X$ (by II, 8.2A). $$\Omega_{K(X)/A} = F/\{\cdots , da | a\in A\} \\ \supset F/\{\cdots, da|a\in K(Y) = \text{Quot}(A)\} = \Omega_{K(X)/K(Y)} = 0$$ where $F$ is the free $K(X)$-module (v.s.) generated by $\{db|b\in K(X)\}$. So I don't see how that helps to show $\Omega_{K(X)/A} = 0$.
What am I missing here?
Thank you for your help.