Bounding Hilbert coefficients of parameter ideals

Let (R,m) be a Noetherian local ring of dimension d>0 and depthR≥d−1. Let Q be a parameter ideal of R. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient ei(Q) under certain assumptions on the depth of associated graded ring G(Q). For 2≤i≤d, we show that (1) ei(Q)≤0 provided depthG(Q)≥d−2 and (2) ei(Q)≥−λR(Hmd−1(R)) provided depthG(Q)≥d−1. It is proved that e3(Q)≤0. Further, we obtain a necessary condition for the vanishing of the last coefficient ed(Q). As a consequence, we characterize the vanishing of e2(Q). Our results generalize [5, Theorem 3.2] and [11, Corollary 4.5].