The dual Hilbert-Samuel function of a Maximal Cohen-Macaulay module

TL;DR

This paper investigates the dual Hilbert-Samuel function associated with a maximal Cohen-Macaulay module over a Cohen-Macaulay local ring, focusing on its polynomial nature and analyzing its first two normalized coefficients.

Contribution

It introduces the dual Hilbert-Samuel function for maximal Cohen-Macaulay modules and studies its first two normalized coefficients, extending understanding of its polynomial properties.

Findings

The dual Hilbert-Samuel function is a polynomial.

The first two normalized coefficients are characterized and analyzed.

Abstract

Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega_R$. Let $I$ be an $\m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $n\longmapsto \ell (\Hom_R(M,{\omega_R}/{I^{n+1} \omega_R}))$ the dual Hilbert-Samuel function of $M$ with respect to $I$. By a result of Theodorescu this function is a polynomial function. We study its first two normalized coefficients.