On a Zeta-Barnes type function associated to graded modules
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866911858266472448 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | 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 $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}ζ_M(z,w)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_11510 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On a Zeta-Barnes type function associated to graded modules Cimpoeas, Mircea Commutative Algebra 13D40, 11M41, 11P81 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 $M$, $e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}ζ_M(z,w)$. |
| title | On a Zeta-Barnes type function associated to graded modules |
| topic | Commutative Algebra 13D40, 11M41, 11P81 |
| url | https://arxiv.org/abs/1803.11510 |