Depth Sensitivity of Hilbert Coefficients

TL;DR

This work investigates how the Hilbert coefficients ${\mathrm e}_{i}(M)$ of a finitely generated graded $R$-module $M$ reflect depth, extending the classical Cohen–Macaulay characterization via multiplicities. It introduces admissible subsystems of parameters (ssop) in $[R]_1$ and proves depth-sensitive inequalities: for $0\le i< s$ with an admissible $ssop$ of length $s-i$, ${\mathrm e}_{i}(M)$ compares to ${\mathrm e}_{i}(M/(f_1,\dots,f_{s-i})M)$ with signs determined by the parity of $i$, and equality occurs exactly when ${\rm depth}_{R}M \ge s-i$. The paper also characterizes admissible ssops as those generated by superficial sequences, provides tools via relative Hilbert coefficients, and demonstrates the approach with explicit computations in graded polynomial quotients. These results offer a practical framework for depth-aware computation of Hilbert coefficients and extend their applicability to understanding minimal reductions and related depth phenomena in graded settings.

Abstract

The purpose of this paper is to explain about the depth sensitivity of the Hilbert coefficients defined for finitely generated graded modules over graded rings. The main result generalize the well known fact that the Cohen-Macaulayness of graded modules can be characterized using their multiplicities.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Example 2.8
• Definition 3.1