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 sets $\Lambda_i^K(R):={g_i^K(Q): Q$ is a parameter ideal of $R},$ where $g_i^K(Q)$ denotes the Hilbert coefficients of $Q$ with respect to $K,$ for $1 \leq i \leq d.$ We prove that $\Lambda_i^K(R)$ is finite for all $1\leq i \leq d$ if and only if $R$ is generalized Cohen-Macaulay. Moreover, we show that if $R$ is unmixed then finiteness of the set $\Lambda_1^K(R)$ suffices to conclude that $R$ is generalized Cohen-Macaulay. We obtain partial results for $R$ to be Buchsbaum in terms of $|\Lambda_i^K(R)|=1.$ We also obtain a criterion for the set $\Delta^K(R):={g_1^K(I): I$ is an m-primary ideal of $R}$ to be finite, generalizing preceding results.