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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2018
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/1803.11510 |
| Etiquetas: |
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- 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 $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}ζ_M(z,w)$.