Hilbert Coefficients and Depth of the Associated Graded Ring of an Ideal

TL;DR

This paper surveys the relationship between Hilbert coefficients of m-primary ideals in Cohen-Macaulay local rings and the depth of their associated graded rings, offering simplified proofs via superficial sequences.

Contribution

It provides accessible proofs of key theorems linking Hilbert coefficients and the depth of associated graded rings, enhancing understanding in commutative algebra.

Findings

Connections between Hilbert coefficients and depth established

Simplified proofs using superficial sequences provided

Results build on theorems of Huckaba and Marley

Abstract

In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences.