Cohen--Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals

TL;DR

This paper proves Vasconcelos's conjecture that the first Hilbert coefficient $e_1(Q)$ vanishes in certain Noetherian local rings and explores its implications for ring properties like Buchsbaum rings.

Contribution

It provides an affirmative proof of Vasconcelos's conjecture and investigates the properties of rings with vanishing $e_1(Q)$, linking it to Buchsbaum rings.

Findings

Vasconcelos's conjecture is proven true.

Properties of rings with $e_1(Q)=0$ are characterized.

Relationship between $e_1(Q)$ and Buchsbaum rings is established.

Abstract

The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and its relationship to Buchsbaum rings are studied.