On the Hilbert coefficients, depth of associated graded rings and\n reduction numbers

2017/03/23 by Амир Мафи, Mafi, Amir, Dler Naderi +1
Mathematics · #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.1703.07961

openalex publication_date 2017/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (R, mathfrakm) be a d-dimensional Cohen-Macaulay local ring, I an\n mathfrakm-primary ideal of R and J=(x₁,...,x_d) a minimal reduction\nof I. We show that if J_d-1=(x₁,...,x_d-1) and\n\∑\_n=1^\∞\λ(Iⁿ⁺¹\∩ J_d-1)/(JIⁿ \∩\nJ_d-1)=i where i=0,1, then depth G(I)\≥d-i-1. Moreover, we prove\nthat if e₂(I) = \∑_n=2^\∞ (n-1) \λ (Iⁿ/JI^n-1)-2; or if I\nis integrally closed and e₂(I) = \∑_n=2^\∞\n(n-1)\λ(Iⁿ/JI^n-1)-i where i=3,4, then e₁(I) =\n\∑_n=1^\∞ \λ(Iⁿ / JI^n-1)-1. In addition, we show that r(I)\nis independent. Furthermore, we study the independence of r(I) with some\nother conditions.\n

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