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On a Zeta-Barnes type function associated to graded modules

Mircea Cimpoeas

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 $ζ_M(z,w):= \sum_{n=0}^{\infty}\frac{H(M,n)}{(n+w)^z}$, 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 $ζ_M(z,w)$. In particular, in the standard graded case, we prove that the multiplicity of $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}ζ_M(z,w)$.

On a Zeta-Barnes type function associated to graded modules

Abstract

Let be a field and let be a positively graded -algebra. Given , a finitely generated graded -module, and , we introduce the function , where , , is the Hilbert function of , and we study the relations between the algebraic properties of and the analytic properties of . In particular, in the standard graded case, we prove that the multiplicity of , .

Paper Structure

This paper contains 4 sections, 15 theorems, 81 equations.

Table of Contents

  1. Graded modules over positively graded $K$-algebras
  2. Graded modules over the ring of polynomials.
  3. The standard graded case
  4. The non-graded case

Key Result

Theorem 1.1

We have that where $\zeta(z,w)=\sum_{n=0}^{\infty}\frac{1}{(n+w)^z}$ is the Hurwitz-zeta function. Moreover, $\zeta_M(z,w)$ is a meromorphic function on $\mathbb C$ with the poles in the set $\{1,2,\ldots,m\}$ which are simple with residues

Theorems & Definitions (33)

  • Theorem 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • ...and 23 more