Bounding Hilbert coefficients of parameter ideals

2016/11/10 by Saikia, Anupam, Saloni, Kumari
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.03431

Abstract

Let (R,\mathfrakm) be a Noetherian local ring of dimension d>0 and depth R≥ 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≤ i≤ d , we show that (1) e_i(Q)≤ 0 provided depth G(Q)≥ d-2 and (2) e_i(Q)≥ -λ_R(H_\mathfrakm^d-1(R)) provided depth G(Q)≥ d-1. It is proved that e₃(Q)≤ 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 e₂(Q). Our results generalize \cite[Theorem 3.2]goto-ozeki and \cite[Corollary 4.5]Lori.

Related

• Hilbert Coefficients of Parameter Ideals
• Hilbert Coefficients and Depth of the Associated Graded Ring of an Ideal
• Restrictions on Hilbert coefficients give depths of graded domains
• Depth of associated graded rings via Hilbert coefficients of ideals
• How to compute the Hilbert depth of a graded ideal