On the finiteness of the set of Hilbert coefficients

Abstract

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].