A note on Ratliff-Rush filtration, reduction number and postulation number of \mathfrak m-primary ideals

2023/07/03 by Mousumi Mandal, Mandal, Mousumi, Shruti Priya +1
Mathematics · Medicine · #13A30 #13D40 #13H10 #Algebraic structures and combinatorial models #Cholinesterase and Neurodegenerative Diseases #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2307.01196

openalex publication_date 2023/07/03 · openalex created_date 2023/07/05 · openalex updated_date 2026/07/28

Abstract

Let (R,\mathfrak m) be a Cohen-Macaulay local ring of dimension d≥ 2 and I an \mathfrak m-primary ideal. Let rd(I) be the reduction number of I and n(I) the postulation number. We prove that for d=2, if n(I)=ρ(I)-1, then rd(I) ≤n(I)+2 and if n(I)≠ ρ(I)-1, then rd(I)≥n(I)+2. For d ≥ 3, if I is integrally closed, depth gr(I) = d-2 and n(I)=-(d-3). Then we prove that rd(I)≥n(I)+d. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to I mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension d≥2, if P_I(k)=H_I(k) for some k ≥ ρ(I), then P_I(n)=H_I(n) for all n ≥ k.

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