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

2023/04/10 by Saloni, Kumari, Yadav, Anoot Kumar
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.04524

Abstract

Let (R,\mathfrakm) be a Cohen-Macaulay local ring of dimension d≥ 3 and I an integrally closed \mathfrakm-primary ideal. We establish bounds for the third Hilbert coefficient e₃(I) in terms of the lower Hilbert coefficients e_i(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 depth G(I)≥ 1, guarantees that Rossi's bound for reduction number r_J(I) holds in dimension three. In that context, we prove that if depth G(I)≥ d-3, then r_J(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).

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