Bounding Hilbert coefficients of parameter ideals

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\ge d-1$. 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 $2\le i\le d$, we show that (1) $e_i(Q)\le 0$ provided depth $G(Q)\ge d-2$ and (2) $e_i(Q)\ge -{\lambda }_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\ge d-1$. It is proved that $e_3(Q)\le 0$. 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}.

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