On a Zeta-Barnes type function associated to graded modules

TL;DR

This paper introduces a new zeta-type function associated with graded modules over graded algebras, linking algebraic invariants like multiplicity to the analytic properties of this function.

Contribution

It defines the zeta_M function for graded modules and establishes a relation between the module's multiplicity and the residue of this function.

Findings

The function $zeta_M(z,w)$ generalizes classical zeta functions for modules.

The multiplicity $e(M)$ is expressed as a limit involving the residue of $zeta_M(z,w)$.

Analytic properties of $zeta_M(z,w)$ reflect algebraic characteristics of the module.

Abstract

Let $K$ be a field and let $S=\bigoplus_{n\geq 0} S_n$ be a positively graded $K$-algebra. Given $M=\bigoplus_{n\geq 0} M_n$, a finitely generated graded $S$-module, and $w>0$, we introduce the function $\zeta_M(z,w):= \sum_{n=0}^{\infty}\frac{H(M,n)}{(n+w)^z}$, where $H(M,n):=\dim_K M_n$, $n\geq 0$, is the Hilbert function of $M$, and we study the relations between the algebraic properties of $M$ and the analytic properties of $\zeta_M(z,w)$. In particular, in the standard graded case, we prove that the multiplicity of $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}\zeta_M(z,w)$.