On the initial Betti numbers
Mohsen Asgharzadeh
TL;DR
This work analyzes the initial Betti numbers of the canonical module $eta_i( abla_R)$ over Cohen–Macaulay local rings with canonical module, establishing general and specialized growth bounds via syzygy ranks and exact sequences. It extends Gover–Ramras-type inequalities to $ abla_R$, derives explicit differences such as $eta_1( abla_R)-eta_0( abla_R)$ in rings like $R=S/ rak n^n$, and explores consequences for semidualizing modules and non-Gorenstein phenomena. The authors apply these bounds to monomial and radical ideals, connect to Dutta–Griffith-type inequalities, and address questions of Huneke and Eisenbud–Herzog by providing lower bounds on the type in products and powers, yielding broad non-Gorenstein criteria. Overall, the results deepen understanding of Betti growth, semidualizing structures, and Gorenstein properties in CM rings, with implications for the structure of canonical modules and monomial ideals.
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 $β_0(ω_{\frac{R}{I}})$ in some cases, where $I$ is a product of two ideals.