On a Zeta-Barnes type function associated to graded modules

Abstract

Let $K$ be a field and let $S={⨁}_{n\ge 0}{S}_n$ be a positively graded $K$-algebra. Given $M={⨁}_{n\ge 0}{M}_n$, 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 $H(M,n):={\dim }_K{M}_n$, $n\ge 0$, 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 $M$, $e(M)=(m-1)!{\lim }_{w\searrow 0}Re{s}_{z=m}{\zeta }_M(z,w)$.

12 pages