Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen–Macaulay modules, which we review in an Appendix.

For an embedding of sufficiently high degree of a smooth projective variety X into projective space, we use residues to define a filtered holonomic D-module (M, F) on the dual projective space. This gives a concrete description of the intermediate extension to a Hodge module on P of the variation of Hodge structure on the middle-dimensional cohomology of the hyperplane sections of X . We also e...