Ideals Whose First Two Betti Numbers are Close

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.