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 $1\le i\le d.$ We prove that ${\Lambda }_i^K(R)$ is finite for all $1\le i\le d$ 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.