Ratliff-Rush filtration, Hilbert coefficients and the reduction number of integrally closed ideals

Abstract

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\ge 3$ and $I$ an integrally closed $\mathfrak{m}$-primary ideal. We establish bounds for the third Hilbert coefficient $e_3(I)$ in terms of the lower Hilbert coefficients $e_i(I),0\le i\le 2$ and the reduction number of $I$. When $d=3$, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of $I$. These properties, though weaker than depth $G(I)\ge 1$, guarantees that Rossi's bound for reduction number $r_J(I)$ holds in dimension three. In that context, we prove that if $depthG(I)\ge d-3$, then $r_J(I)\le e_1(I)-e_0(I)+\ell (R/I)+1+e_2(I)(e_2(I)-e_1(I)+e_0(I)-\ell (R/I))-e_3(I).$ We also discuss the signature of the fourth Hilbert coefficient $e_4(I).$

Comments are welcome