Upper bounds on two Hilbert coefficients

TL;DR

This paper establishes new upper bounds for the first two Hilbert coefficients of Cohen-Macaulay modules over local rings, providing characterizations for when these bounds are achieved using Hilbert series and Castelnuovo-Mumford regularity.

Contribution

It introduces novel upper bounds for Hilbert coefficients and characterizes the cases of equality through Hilbert series and regularity measures.

Findings

New upper bounds for Hilbert coefficients established

Characterizations of bounds achieved via Hilbert series

Connections made with Castelnuovo-Mumford regularity

Abstract

New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of Hilbert series as well as in terms of the Castelnuovo-Mumford regularity of the associated graded module.