# Dependence of Hilbert coefficients

## TL;DR

This paper investigates the relationships among Hilbert coefficients of modules over Noetherian local rings, establishing bounds for the last coefficients based on the initial ones and the module's dimension.

## Contribution

It provides new bounds for the last Hilbert coefficients in terms of the initial coefficients and the module's dimension, enhancing understanding of their dependence.

## Key findings

- Last $t$ Hilbert coefficients are bounded by the first $d-t+1$ coefficients.
- The bounds depend explicitly on the module's dimension $d$.
- Results apply to modules of arbitrary depth over Noetherian local rings.

## Abstract

Let $M$ be a finitely generated module of dimension $d$ and depth $t$ over a Noetherian local ring ($A, {\mathfrak m}$) and $I$ an ${\mathfrak m}$-primary ideal. In the main result it is shown that the last $t$ Hilbert coefficients $e_{d-t+1}(I,M),..., e_d(I,M)$ are bounded below and above in terms of the first $d-t+1$ Hilbert coefficients $e_0(I,M),...,e_{d-t}(I,M)$ and $d$.