On the cohomology of regular differential forms and dualizing sheaves

If $Y$ is a local Dedekind scheme and $X/Y$ is a projective Cohen-Macaulay variety of relative dimension $1$, then $R^{1} f_{*} \omega ^{1}_{X/Y}$ is torsionfree if and only if $X/Y$ is arithmetically Cohen-Macaulay for a suitable embedding in $\mathbb{P}^{n}_{k}$. If $X$ is regular then $R^{1} f_{*} \omega ^{1}_{X/Y}$ is torsionfree whenever the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field.