An Upper Bound for the First Hilbert Coefficient of Gorenstein Algebras and Modules

TL;DR

This paper establishes an upper bound for the first Hilbert coefficient of Gorenstein modules and algebras based on their graded resolutions, extending previous bounds and proposing a conjecture for higher coefficients.

Contribution

It provides a new upper bound for the first Hilbert coefficient of Gorenstein modules and algebras using the shifts in their minimal free resolutions, generalizing earlier results.

Findings

Upper bound for e_1 in Gorenstein modules derived

Bound matches previous results for Gorenstein algebras with quasi-pure resolutions

Conjecture proposed for bounds on higher Hilbert coefficients

Abstract

Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay module $M$ is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, $e_1$ in terms of the shifts in the graded resolution of $M$. When $M = R/I$, a Gorenstein algebra, this bound agrees with the bound obtained in \cite{ES} in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.