On the finiteness of the set of Hilbert coefficients

Abstract

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be $\mathfrak m$-primary ideals in $R$. In this paper we study the finiteness properties of the sets $\Lambda _i^K(R):=\{g_i^K(Q): Q\mbox { is a parameter ideal of }R\}$, 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 $|\Lambda _i^K(R)|=1$. Our results are more general than previous work of Goto, Ozeki, and others. We also obtain a criterion for the set $\Delta ^K(R):=\{g_1^K(I): I\mbox { is an $\mathfrak m$-primary ideal of }R\}$ to be finite, generalizing a result of Koura and Taniguchi.