On Hilbert coefficients of parameter ideals and Cohen-Macaulayness

TL;DR

This paper establishes a criterion for Cohen-Macaulayness of unmixed local rings based on Hilbert coefficients of parameter ideals, confirming a conjecture and advancing understanding of ring properties.

Contribution

It provides a necessary and sufficient condition for Cohen-Macaulayness using Hilbert coefficients, and confirms Vasconcelos's negativity conjecture in this context.

Findings

Characterization of Cohen-Macaulay rings via Hilbert coefficients

Resolution of Vasconcelos's negativity conjecture for unmixed rings

Link between Hilbert coefficients and ring properties

Abstract

Let $(R, \mathfrak m)$ be an unmixed Noetherian local ring, Q a parameter ideal and $K$ an $\mathfrak m$-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and $g_1(Q)$, the Hilbert coefficients of $Q$ with respect to $K$. As a consequence, we obtain a result of Ghezzi et al. which settles the negativity conjecture of W. V. Vasconcelos [15] in unmixed local rings.