Abstract. Let (R, m) 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 ei(Q) under certain assumptions on the depth of associated graded ringFurther, 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

Let (R, m) be a Cohen-Macaulay local ring of dimension d ≥ 3 and I an integrally closed m-primary ideal. We establish bounds for the third Hilbert coefficient e 3 (I) in terms of the lower Hilbert coefficients e i (I), 0 ≤ i ≤ 2 and the reduction number of I. When d = 3, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of I. These properties, though weaker than depth G(I) ≥ 1, guarantees that Rossi's bound for reduction number r J (I) holds in dimension three. In that context, we prove that if depth G(I) ≥ d − 3, then r J (I) ≤ e 1 (I) − e 0 (I) + ℓ(R I) + 1 + e 2 (I)(e 2 (I) − e 1 (I) + e 0 (I) − ℓ(R I)) − e 3 (I). We also discuss the signature of the fourth Hilbert coefficient e 4 (I).

Let (R, m) be an unmixed Noetherian local ring, Q a parameter ideal and K an m-primary ideal of R containing Q. We give a necessary and sufficient condition for R to be Cohen-Macaulay in terms of g0(Q) and g1(Q), the Hilbert coefficients of Q with respect to K. As a consequence, we obtain a result of Ghezzi et al. which settles the negativity conjecture of W. V. Vasconcelos [15] in unmixed local rings.

Let (R, m) be a Noetherian local ring of dimension d and K, Q be m-primary ideals in R. In this paper we study the finiteness properties of the setsMoreover, we show that if R is unmixed then finiteness of the set Λ K 1 (R) suffices to conclude that R is generalized Cohen-Macaulay. We obtain partial results for R to be Buchsbaum in terms of |Λ K i (R)| = 1. Our results are more general than in [GGH + 15] and [GO11]. We also obtain a criterion for the set ∆ K (R) := {g K 1 (I) : I is an m-primary ideal of R} to be finite, generalizing a result of [KT15].