Bezout's theorem and Cohen-Macaulay modules

We define very proper intersections of modules and projective subschemes. I t turns out that equidimensional locally Cohen-Macaulay modules intersect v ery properly if and only if they intersect properly. We prove a Bezout theo rem for modules which meet very properly. Furthermore, we show for equidime nsional subschemes X and Y: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then X and Y are arithmetic ally Cohen-Macaulay. The module version of this result implies splitting cr iteria for reflexive sheaves.