arXiv math/99120661999-12-08math.AGmath.APmath.RA

特征簇的不可约分支

Irreducible components of characteristic varieties

Gregory G. Smith

备注：: 20 pages
发表信息：: Journal of Pure and Applied Algebra 165 (2001) 291-306
DOI：: 10.1016/S0022-4049(01)00109-8

摘要：

本文给出了Weyl代数$A_{n}(k)$的一个适当滤子所定义的线性偏微分方程组的特征簇的不可约分量的维数界。这推广了一个重要的事实，即从过滤的顺序定义的特征品种是对合的后果。更明确地说，我们考虑了由任何向量$(u,v) \in {\mathbb Z}^{n}\times {\mathbb Z}^{n}$诱导的$A_{n}(k)$的过滤，使得相关的分次代数是$2n$不定式中的交换多项式环。阶次滤波是$(u,v) = (0,1)$的特例。任何反向生成的左$A_{n}(k)$ -模块$M$相对于$(u,v)$具有良好的过滤，并且这产生了仅依赖于$(u,v)$和$M$的特征变体$\Ch_{(u,v)}(M)$。当$(u,v) = (0,1)$时，特征簇是对合的，这意味着它的不可约分支的维数至少为$n$。一般来说，特征簇可能不是对合的，但我们仍然能够证明$\Ch_{(u,v)}(M)$的每个不可约分支至少有$n$维数。

英文摘要：

We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial differential equations defined from a suitable filtration of the Weyl algebra $A_{n}(k)$. This generalizes an important consequence of the fact that a characteristic variety defined from the order filtration is involutive. More explicitly, we consider a filtration of $A_{n}(k)$ induced by any vector $(u,v) \in {\mathbb Z}^{n}\times {\mathbb Z}^{n}$ such that the associated graded algebra is the commutative polynomial ring in $2n$ indeterminates. The order filtration is the special case $(u,v) = (0,1)$. Any finitely generated left $A_{n}(k)$-module $M$ has a good filtration with respect to $(u,v)$ and this gives rise to a characteristic variety $\Ch_{(u,v)}(M)$ which depends only on $(u,v)$ and $M$. When $(u,v) = (0,1)$, the characteristic variety is involutive and this implies that its irreducible components have dimension at least $n$. In general, the characteristic variety may fail to be involutive, but we are still able to prove that each irreducible component of $\Ch_{(u,v)}(M)$ has dimension at least $n$.

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