Stable polynomial division and essential normality of graded Hilbert modules
Stable polynomial division and essential normality of graded Hilbert modules
复制标题
DOI：
10.1112/jlms/jdq054
复制
发表时间：
2010-03
期刊：
影响因子：
--
通讯作者：
O. Shalit
中科院分区：
文献类型：
--
作者：
O. Shalit
本文的目的是对Arveson的抵抗猜想发起一个新的攻击，即d移位Hilbert模H2的所有分次子模本质正规。我们引入了模（和理想）的稳定除性质：d元多项式环上的赋范模M具有稳定除性质，如果它有一个生成集{f1，...，fk}使得每个h∈M可以写为h = ∑i ai fi，对于某些多项式ai，使得∑ i ai fi C。我们证明了某些类的模具有这种性质，并且可以通过仔细应用计算代数的算法来获得稳定分解h = ∑ ai fi。证明了当d元多项式代数具有自然范数时，每个理想都线性等价于一个具有稳定除法性质的理想。然后我们证明了，对于一个具有稳定除性质的子模M（关于适当的范数），商模H2/M对于p > dim（M）是p-本质正规的，正如道格拉斯所证明的。这个结果被用来给出某些分次子模类本质正规的新的统一证明。最后，我们将确定d移位希尔伯特模的所有分次子模是否本质正规的问题归结为确定由二次标值多项式生成的所有理想是否本质正规的问题。
The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the d‐shift Hilbert module H2 are essentially normal. We introduce the stable division property for modules (and ideals): a normed module M over the ring of polynomials in d variables has the stable division property if it has a generating set {f1, …, fk} such that every h∈M can be written as h = ∑i ai fi for some polynomials ai such that ∑ ‖ai fi‖ ⩽ C‖h‖. We show that certain classes of modules have this property, and that the stable decomposition h = ∑ ai fi may be obtained by carefully applying algorithms from computational algebra. We show that when the algebra of polynomials in d variables is given the natural ℓ1 norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that, for a submodule M that has the stable division property (with respect to the appropriate norm), the quotient module H2/M is p‐essentially normal for p > dim(M), as conjectured by Douglas. This result is used to give a new unified proof that certain classes of graded submodules are essentially normal. Finally, we reduce the problem of determining whether all graded submodules of the d‐shift Hilbert module are essentially normal, to the problem of determining whether all ideals generated by quadratic scalar‐valued polynomials are essentially normal.