Remarks on Hilbert series of graded modules over polynomial rings
Remarks on Hilbert series of graded modules over polynomial rings
复制标题
关于多项式环上希尔伯特级数模的备注
DOI：
10.1007/s00229-010-0341-9
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发表时间：
2010
影响因子：
0.6
通讯作者：
Jan Uliczka
中科院分区：
文献类型：
--
作者：
Jan Uliczka
本文讨论了标准分次多项式环上的形式Laurent级数的一个结果及其对Hilbert生成分次模级数的一些意义：对于任何具有非负值的多项式类型的整数Laurent函数，相关的形式Laurent级数可以写成形式为$${\frac{Q_j（t）}{（1-t）^j}}$$的有理函数的和，其中分子是具有非负整数系数的劳伦多项式。因此，任何这样的级数都是一个合适的多项式环$${\mathbb{F}[X_1，\ldots，X_n]}$$上的某个生成分次模的希尔伯特级数。我们给出了两个进一步的应用，即研究具有给定希尔伯特级数的模的最大深度，以及在将希尔伯特级数表示为幂为（1 − t）的有理函数作为分母时，可能作为分子出现的洛朗多项式的特征。
In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form $${\frac{Q_j(t)}{(1-t)^j}}$$, where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring $${\mathbb{F}[X_1 , \ldots , X_n]}$$. We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 − t) as denominator.