Bounding reduction number and the Hilbert coefficients of filtration

2024/09/23 by Kumari Saloni, Saloni, Kumari, Anoot Kumar Yadav +1
Mathematics · #13A30 #13D40 #13H10 #Advanced Optimization Algorithms Research #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2409.14860

openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (A,\m) be a Cohen-Macaulay local ring of dimension d≥ 3, I an \m-primary ideal and I=\I_n\_{n≥ 0} an I-admissible filtration. We establish bounds for the third Hilbert coefficient: (i) e₃(I)≤ e₂(I)(e₂(I)-1) and (ii) e₃(I)≤ e₂(I)(e₂(I)-e₁(I)+e₀(I)-ℓ(A/I)) if I is an integrally closed ideal. Further, assume the respective boundary cases along with the vanishing of e_i(I) for 4≤ i≤ d. Then we show that the associated graded ring of the Ratliff-Rush filtration of I is almost Cohen-Macaulay, Rossi's bound for the reduction number r_J(I) of I holds true and the reduction number of Ratliff-Rush filtration of I is bounded above by r_J(\I). In addition, if \wtI^r_J(I)=I^r_J(I), then we prove that \reg G_I(A)=r_J(I) and a bound on the stability index of Ratliff-Rush filtration is obtained. We also do a parallel discussion on the \textquotedblleft good behaviour of the Ratliff-Rush filtration with respect to superficial sequence''.

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