Bounding Hilbert coefficients of parameter ideals

TL;DR

This paper establishes bounds for Hilbert coefficients of parameter ideals in certain Noetherian local rings, linking these bounds to the depth of associated graded rings and providing characterizations for the vanishing of specific coefficients.

Contribution

It introduces new uniform bounds for Hilbert coefficients based on depth conditions and characterizes when these coefficients vanish, extending previous results.

Findings

Bounded $e_i(Q)$ for $2 \\leq i \\leq d$ under depth conditions.

Proved $e_3(Q) \\leq 0$.

Characterized vanishing of $e_2(Q)$ and conditions for $e_d(Q)=0$.

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain assumptions on the depth of associated graded ring $G(Q)$. For $2\leq i\leq d $, we show that (1) $e_i(Q)\leq 0$ provided depth $G(Q)\geq d-2$ and (2) $e_i(Q)\geq -\lambda_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\geq d-1$. It is proved that $e_3(Q)\leq 0$. Further, we obtain a necessary condition for the vanishing of the last coefficient $e_d(Q)$. As a consequence, we characterize the vanishing of $e_2(Q)$. Our results generalize \cite[Theorem 3.2]{goto-ozeki} and \cite[Corollary 4.5]{Lori}.