# On the finiteness of the set of Hilbert coefficients

## TL;DR

This paper investigates the finiteness of sets of Hilbert coefficients in Noetherian local rings, establishing conditions under which these sets are finite and characterizing the ring's properties such as being generalized Cohen-Macaulay or Buchsbaum.

## Contribution

It provides a characterization of when the sets of Hilbert coefficients are finite, linking this to the ring being generalized Cohen-Macaulay or Buchsbaum, and extends previous results with new criteria.

## Key findings

- Finiteness of Hilbert coefficient sets characterizes generalized Cohen-Macaulay rings.
- Unmixed rings with finite first Hilbert coefficient set are generalized Cohen-Macaulay.
- Partial results relate the size of coefficient sets to Buchsbaum properties.

## 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.