KRULL DIMENSION OF MODULES AND INVOLUTIVE IDEALS
KRULL DIMENSION OF MODULES AND INVOLUTIVE IDEALS
复制标题
模块的克鲁尔维度和包容性理想
DOI：
10.1090/s0002-9939-1995-1243163-9
复制
发表时间：
1995
影响因子：
6.7
通讯作者：
S. C. Coutinho
中科院分区：
文献类型：
--
作者：
S. C. Coutinho
本文利用Weyl代数的特征簇的一个几何不变量--对合维数，建立了Weyl代数上模的Krull维数的一个上界.这之后的一些例子表明，这个不等式可能是严格的。本文证明了Weyl代数上模的Krull维数与该模的特征簇的几何不变量有关。我们开始调查的一些基本事实的模在Weyl代数和他们的特征品种。设An表示复数域C上的第n阶Weyl代数。代数An的生成元记为xi，di = d/dxi，其中I gh} - g{f，h} + h{f，g}。它可以用来计算A”中两个算子的括号的符号如下：如果d ∈ A”（k）和d'∈ An（m），则ak+m_2（（d，d'））= {ak（d），am（d '）}。更多细节见（1）或（10）。设M是一个生成的左？（“-模和F是M的一个很好的过滤。通过这一点，我们意味着M的过滤，其中%rF M是在Sn上连续生成的。设I（M）是grf M在S”中的零化子的根.注意I（M）是S”的齐次理想。理想I（M）称为M的特征理想，其簇char（AZ）在C2”中称为M的特征簇。I（M）和char（Af）都与良好过滤的选择无关
In this paper we establish an upper bound for the Krull dimension of a module over a Weyl algebra in terms of a geometrical invariant of its char- acteristic variety, the involutive dimension. This is followed by some examples which show that this inequality may be strict. In this paper we show that the Krull dimension of a module over the Weyl algebra is related to geometrical invariants of the characteristic variety of that module. We begin with a survey of a few basic facts about modules over the Weyl algebra and their characteristic varieties. Let An denote the nth Weyl algebra over the field of complex numbers C. The generators of the algebra An will be denoted by xi and di = d/dxi for I gh} — g{f, h} + h{f, g}. It may be used to calculate the symbol of a bracket of two operators in A" as follows: If d £ A"(k) and d' £ An(m), then ak+m_2((d, d')) = {ak(d), am(d')}. For more details see (1) or (10). Let M be a finitely generated left ?("-module and F a good filtration of M. By that we mean a filtration of M for which %rF M is finitely generated over Sn . Let I(M) be the radical of the annihilator of grf M in S" . Note that I(M) is a homogeneous ideal of S" . The ideal I(M) is called the characteristic ideal of M and its variety char(AZ) in C2" , the characteristic variety of M. Both I(M) and char(Af ) are independent of the choice of the good filtration