Ideals Whose First Two Betti Numbers are Close

TL;DR

This paper investigates the relationship between the first two Betti numbers of ideals in Noetherian local rings, establishing bounds and exploring their implications for residual intersections and canonical modules.

Contribution

It proves a lower bound on the difference of the first two Betti numbers and characterizes residual intersections with minimal Betti number differences as perfect.

Findings

Proves that ^R(I) - \u00b0^R(I) a9 -1 for ideals in Noetherian local rings.

Shows residual intersections with Betti number differences of -1 or 0 are perfect.

Explores relations between Betti numbers and Bass numbers of the canonical module.

Abstract

For an ideal $I$ of a Noetherian local ring $(R,\fm,k)$ we show that $\bt_1^R(I)-\bt_0^R(I)\geq -1$. It is demonstrated that some residual intersections of an ideal $I$ for which $\bt_1^R(I)-\bt_0^R(I)= -1\;\text{or}\;0$ are perfect. Some relations between Betti numbers and Bass numbers of the canonical module are studied.