On the initial Betti numbers

Abstract

Let $R$ be a Cohen-Macaulay local ring possessing a canonical module. We compare the initial and terminal Betti numbers of modules in a series of nontrivial cases. We pay special attention to the Betti numbers of the canonical module. Also, we compute ${\hat{I}}_0^2(\ddot{I}{‰}_{\frac{R}{I}})$ in some cases, where $I$ is a product of two ideals.