Let $X/S$ be a proper, smooth relative curve over a Dedekind scheme $S$, for example, $X = \mathbb{P}^1_S \xrightarrow{\pi} S$.
Suppose that $Z \to X$ is a horizontal effective Cartier divisor such that $Z/S$ is a finite extension of Dedekind schemes.
If we take the twisted ideal sequence $$0 \to \mathcal{O}_X \to \mathcal{O}_X(Z) \to \mathcal{O}_Z(Z) \to 0$$ and tensor with the sheaf $\Omega_{X/S}$ of differentials, we obtain $$0 \to \Omega_{X/S} \to \Omega_{X/S}(Z) \to \Omega_{X/S}(Z)|_Z \to 0$$
Two of these sheaves are dualizing sheaves, so we can rewrite as $$0 \to \omega_{X/S} \to \omega_{X/S}(Z) \to \omega_{Z/X} \to 0.$$
Taking the long exact sequence for $\pi_*$, we get $$0 \to \pi_*\Omega_{X/S}(Z) \to \pi_*\Omega_{X/S}(Z)|_Z \to R^1\pi_*\Omega_{X/S} \simeq \mathcal{O}_S \to 0.$$
Since $R^0\pi_*\Omega_{X/S} = 0$, and the dualizing sheaf $\Omega_{X/S} \simeq \omega_{X/S}$ has a canonical trace map to $\mathcal{O}_S$.
Question 1: I know that $R^1\pi_*\Omega_{X/S} \simeq \mathcal{O}_S$, but how do I analyze the terms $\pi_*\Omega_{X/S}(Z)$ and $\pi_*\Omega_{X/S}|_Z$? I am trying to obtain a concrete description of them.
Question 2: We also know $\omega_{Z/X}$ is a line bundle on $Z$, corresponding to an element of the ideal class group of the Dedekind scheme $Z$. How can I determine the isomorphism class of this line bundle, or whether it is trivial?
Thank you.
