Vanishing and non-negativity of the first normal Hilbert coefficient

TL;DR

This paper investigates conditions under which the first normal Hilbert coefficient vanishes or is non-negative in Noetherian local rings, establishing regularity criteria and extending previous results using big Cohen-Macaulay algebras.

Contribution

It proves that vanishing of the first normal Hilbert coefficient implies regularity under certain conditions and provides a strengthened proof of related results, advancing understanding of Hilbert coefficients.

Findings

Vanishing of _1(Q) implies R is regular.

_1(Q)  0 for all parameter ideals Q in equidimensional rings.

In characteristic p>0, _1^*(Q)  0 for all parameter ideals Q.

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and $\nu(\mathfrak{m}/Q)\leq 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\geq 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^*(Q)\geq 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras.