Restrictions on Hilbert coefficients give depths of graded domains

2025/01/14 by Cheng Meng, Meng, Cheng
Mathematics · #13P10 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Commutative Algebra (math.AC) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2501.07829

openalex publication_date 2025/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that if P is a homogeneous prime ideal inside a standard graded polynomial ring S with dim(S/P)=d, and for s ≤ d, adjoining s general linear forms to the prime ideal changes the (d-s)-th Hilbert coefficient by 1, then depth(S/P)=s-1. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.

Related

• Hilbert Coefficients and Depth of the Associated Graded Ring of an Ideal
• How to compute the Hilbert depth of a graded ideal
• Depth of associated graded rings via Hilbert coefficients of ideals
• Bounding Hilbert coefficients of parameter ideals
• Hilbert coefficients and depths of form rings