Negativity Conjecture for the First Hilbert Coefficient

TL;DR

This paper provides an alternative proof demonstrating that in an unmixed Noetherian local ring, the first Hilbert coefficient of parameter ideals is negative unless the ring is Cohen--Macaulay.

Contribution

It offers a new proof of a known theorem relating the negativity of the first Hilbert coefficient to Cohen--Macaulayness.

Findings

First Hilbert coefficient is negative in non-Cohen--Macaulay rings.

Equality to zero characterizes Cohen--Macaulay rings.

Provides an alternative proof to existing results.

Abstract

This gives an alternate proof of the Theorem by the authors that shows the first Hilbert coefficient of parameter ideals in an unmixed Noetherian local ring is always negative unless the ring is Cohen--Macaulay.