Bezout's theorem and Cohen-Macaulay modules (Q5947793)

Consider two subschemes \(X, Y\subset \mathbb P^n\). The authors define: \(X,Y\) meet very properly if they meet properly and also their cohomology modules do. The main results in this paper are: Theorem. Let \(X, Y\subset \mathbb P^n\) denote equidimensional subschemes which meet very properly in an arithmetically Cohen-Macaulay subscheme of positive dimension. Then \(X\) and \(Y\) are arithmetically Cohen-Macaulay. Theorem. Let \(X, Y\subset \mathbb P^n\) denote equidimensional subschemes wich intersect very properly in a non empty scheme. Then the intersection is equidimensional, and we have \(\deg X\cap Y=\deg X\cdot \deg Y\). Remark that meet properly is not sufficient in the above theorems.