On the initial Betti numbers

TL;DR

This paper investigates the initial Betti numbers of modules over Cohen-Macaulay local rings with canonical modules, highlighting their behavior and computing specific Betti numbers in cases involving ideal products.

Contribution

It provides new comparisons between initial and terminal Betti numbers and computes Betti numbers of canonical modules in specific algebraic settings.

Findings

Initial Betti numbers can differ significantly from terminal Betti numbers.

Explicit formulas for Betti numbers of canonical modules are derived in certain cases.

The study offers insights into the structure of modules over Cohen-Macaulay rings.

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 $\beta_0(\omega_{\frac{R}{I}})$ in some cases, where $I$ is a product of two ideals.