On the cohomology of regular differential forms and dualizing sheaves

Abstract

If Y Y is a local Dedekind scheme and X / Y X/Y is a projective Cohen–Macaulay variety of relative dimension 1 1 , then R 1 f ∗ ω X / Y 1 R^{1} f_{*} \omega ^{1}_{X/Y} is torsionfree if and only if X / Y X/Y is arithmetically Cohen–Macaulay for a suitable embedding in P k n \mathbb {P}^{n}_{k} . If X X is regular then R 1 f ∗ ω X / Y 1 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.