On the finiteness of the set of Hilbert coefficients

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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 sets Λᵢᵏ(R):={gᵢᵏ(Q): Q is a parameter ideal of R}, where gᵢᵏ(Q) denotes the Hilbert coefficients of Q with respect to K, for 1 ≤ i ≤ d. We prove that Λᵢᵏ(R) is finite for all 1≤ i ≤ d if and only if R is generalized Cohen-Macaulay. Moreover, we show that if R is unmixed then finiteness of the set Λ₁ᵏ(R) suffices to conclude that R is generalized Cohen-Macaulay. We obtain partial results for R to be Buchsbaum in terms of |Λᵢᵏ(R)|=1. We also obtain a criterion for the set Δᵏ(R):={g₁ᵏ(I): I is an m-primary ideal of R} to be finite, generalizing preceding results.