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

Abstract

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥3 and I an integrally closed m-primary ideal. We establish bounds for the third Hilbert coefficient e3(I) in terms of the lower Hilbert coefficients ei(I),0≤i≤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 depthG(I)≥1, guarantee that Rossi's bound for reduction number rJ(I) holds in dimension three. In that context, we prove that if depthG(I)≥d−3, then rJ(I)≤e1(I)−e0(I)+ℓ(R/I)+1+e2(I)(e2(I)−e1(I)+e0(I)−ℓ(R/I))−e3(I). We also discuss the signature of the fourth Hilbert coefficient e4(I).