Suppose that $X$ is an arithmetic surface, i.e. $\pi: X \to S$ flat and relative dimension 1 over a Dedekind scheme $S$, and assume $X$ smooth.
Let $Y \subset X$ be a horizontal effective Cartier divisor.
As in https://stacks.math.columbia.edu/tag/0FMU, we have the sheaves of differentials $$\Omega_{X/S}^p \subset \Omega_{X/S}^p(\log Y) \subset \Omega_{X/S}^p(Y),$$
where $\Omega_{X/S}(\log Y)$ is generated by $\Omega_{X/S}$ and elements $\mathrm{dlog}(f) = \frac{\mathrm{d}f}{f}$ for local equations $f$ defining $Y$.
$Y/S$ is a finite flat extension, possible ramified, so not necessarily smooth or having normal crossing singularities.
(1) Do we still have the following short exact sequence of complexes?
$$0 \to \Omega_{X/S}^\bullet \to \Omega_{X/S}^\bullet(\log Y) \to \Omega_{Y/S}^\bullet[-1] \to 0$$
If not, what is the cokernel?
Would it still hold if $X$ is regular with normal crossing singularities?
(2) If $X/S$ is smooth, so that $\Omega_{X/S}$ is locally free, is $\Omega_{X/S}(\log Y)$ also locally free?
Addendum: From Dolgachev's article (p3), if the base $S=k$ is a field of characteristic zero, then we get the exact sequence $$0 \to \Omega^1_{X/k} \to \Omega^1_{X/k}(\log Y) \to \mathcal{E}xt^1_X(\mathcal{J}_Y(Y),\mathcal{O}_X) \to 0$$ where $\mathcal{J}_Y$ is the Jacobian ideal of $Y$ (generated locally by $f,f'$). The ideal $\mathcal{J}_Y$ defines a closed subscheme $Y^s$ supported on the singular locus of $Y$ (i.e. ramification locus in my case), and we have $\mathcal{O}_{Y^s} = \mathcal{O}_X/\mathcal{J}_Y$.
Then some more analysis gives $$0 \to \mathcal{O}_Y \to \mathcal{E}xt^1_X(\mathcal{J}_Y(Y),\mathcal{O}_X) \to \mathcal{E}xt^2_X(\mathcal{O}_{Y^s},\mathcal{O}_X) \to 0.$$ Apparently it is known that the $\mathcal{E}xt^q_X(\mathcal{F},\mathcal{O}_X)$ will vanish if $q<c$ and $\mathcal{F}$ is supported on a closed subset of codimension $c$. Unfortunately in my case the codimension of the singular locus will be $2$, so I can't draw any immediate conclusions.
The reframing in terms of Ext sheaves provides more information, but unfortunately I still have some problems: (a) The results stated by Dolgachev are only over a field, and I am unsure whether they work over a Dedekind domain $S$, (b) I don't know enough about Ext sheaves to gain anything from this analysis.
I would appreciate any help or insight here.
