I am trying to understand the sheaf of relative differentials for the case of nonsingular curves. Let's use Hartshorne as a reference, thus a curve is an integral scheme of dimension 1, proper over $k$, all of whose local rings are regular.
Based on the definition of the curve, and Theorem 8.15 at p. 177, i see that if $X$ is a curve, then $\Omega_{X/k}=\Omega_X$ is a locally free $O_{X}$-module of rank $1$. Now in page 300, it is mentioned that if $u$ is a local parameter at $P \in X$, then $du$ is a generator of the free $O_P$-module $\Omega_{X,P}$.
Could somebody please explain:
1) How does it follow from the fact that $\Omega_{X}$ is a locally free $O_X$-module of rank $1$, that the stalk $\Omega_{X,P}$ is a free $O_{X,P}$ module of rank $1$? 2) I understand that $d$ is some universal derivation; is it the universal $O_{X,P}$ derivation corresponding to the $O_{X,P}$-module $\Omega_{X,P}$? 3) Why is $\Omega_{X,P}$ generated by $du$?
As it might be obvious, i am completely missing the picture here.
Thanks.