Integrally closed ideals of reduction number three

2021/05/15 by Shinya Kumashiro, Kumashiro, Shinya
Mathematics · #13A30 #13D40 #13H10 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2105.07186

openalex publication_date 2021/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a Cohen-Macaulay local ring (A, \mathfrakm), we study the Hilbert function of an integrally closed \mathfrakm-primary ideal I whose reduction number is three. With a mild assumption we give an inequality ℓ_A(A/I) ≥ e₀(I) - e₁(I) + \dfrace₂(I) + ℓ_A(I²/QI)2, where e_i(I) denotes the ith Hilbert coefficients and Q denotes a minimal reduction of I. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of I is larger than or equal to dim A-1. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

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