On a smooth curve $X/k$, at every point $x\in X$ there is a local parameter $z$ such that $\Omega_{X/k}^1 = \langle \mathrm{d}z \rangle$. We may choose any generator of the maximal ideal of the regular local ring $z\in \mathfrak{m}_x \subset \mathcal{O}_{X,x}$ . Does this also work on a relative curve $X/S$? In other words:
Let $\pi:X\to S$ be a smooth relative curve. Then the sheaf of differentials $\Omega_{X/S}^1$ is a locally free $\mathcal{O}_X$-module of rank 1. Given a point $x\in X$, $\Omega_{X/S,x}^1$ is also a locally free $\mathcal{O}_{X,x}$module of rank 1. If $U \ni x$ is an affine open neighborhood, is there an element $z\in \mathcal{O}_X(U)$ such that $\Omega_{X/S}^1(U)$ is generated by $\mathrm{d}z$?
Can we always take $z$ which defines a closed subscheme $Z \ni x$, such that $Z$ is flat over $S$? %By this I mean given an affine $V \supset \pi(U)$, is $\mathcal{O}_X(U)/(z)$ a flat $\mathcal{O}_S(V)$-algebra?
Although $\Omega_{X/S}^1$ is locally principally generated, I realized I wasn't sure how to show that the generator could always be taken to be of the form $\mathrm{d}z$, and not just some $\sum a_i\mathrm{d}z_i$.