On Hilbert coefficients and sequentially Cohen-Macaulay rings

TL;DR

This paper investigates the relationship between Hilbert coefficients and the structure of local rings, providing characterizations of sequentially Cohen-Macaulay, Gorenstein, and Cohen-Macaulay rings through these coefficients.

Contribution

It offers a new characterization of sequentially Cohen-Macaulay rings using Hilbert coefficients for non-parameter ideals, extending understanding of ring properties.

Findings

Characterization of sequentially Cohen-Macaulay rings via Hilbert coefficients.

Characterizations of Gorenstein and Cohen-Macaulay rings using Chern coefficients.

Establishment of links between the index of reducibility and Hilbert coefficients.

Abstract

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients for non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients for parameter ideals.