权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Structure of Rings of Differential Operators and Poisson Algebras

微分算子和泊松代数环的结构

基本信息

批准号：
11640029
负责人：
HIRANO Yasuyuki
金额：
$ 1.47万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640029/
关键词：
Differential Operator Skew Differential Operator Ordered monoid Monoid Ring Shew Polynomial Ring Crossed Product Azumaya Algebra Higher Derivation 順序群歪群環純非分離拡大導分イデアル自己同型入射加群多項式環不変部分環

项目摘要

1. Let A be an affine domain over a field k of characteristic O, and g a k-automorphism of A of order m. We studied the ring D (A ; g) of differential operators introduced by A.D.Bell. We proved that if A is a free module over the fixed subring A' of A by g with a basis containing 1, then D (A ; g) is isomorphic to the ring of m by m matrices over D (A'). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c of A such that D(A[1/C] ; g) is isomorphic to the ring of m by m matrices over D ((A [1/c])). As an application, we determined the structure of D(A ; g) when A is a polynonmial or Laurent polynomial ring over k and g is a diagonalizable linear automorphism.2. A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. We showed that if R is (left principally) quasi-Baer and G is an ordered monoid, then the monoid ring RG is again (left principally) quasi- … More Baer. When R is (left principally) quasi-Baer and G is an ordered group acting on R, we gave a necessary and sufficient condition for the skew group ring R#G to be (left principally) quasi-Baer.3. A ring R is said to be reduced if it has no nonzero nilpotent elements. Let R be a ring and let g be an automorphism of R.We gave a necessaryand sufficient condition for the skew polynomial ring R [x ; g] to be reduced, We also gave some sufficient condition for R [x ; g] to a Baer ring, a quasi-Baer ring, or a principally quasi-Baer ring.4. Let B be a Z-Azumaya algebra, D a derivation, and Z' the element z of Z such that D (z)=0. We proved that if Z/Z' is a purely inseparable extension of exponent 1 and B satisfies some other conditions then the skew polynomial ring B [x ; D] is an Azumaya algebra. Let p be a prime, G a p-group, B a ring, and f a factor set of B.We showed tha if the crossed product Δ (B, G, f) is separable over B and if p is contained in the Jacobson radiucal of B then Δ (B, G, f)/B is an H-separable extension.5. We introduced the notion of generalized higher derivations and developed the relations between them and ordinary higher derivations. We also investigated the categorical properties of generalized higher derivations. Less

1.设A是特征为O的域k上的仿射整环，g是A的m阶k-自同构，我们研究了由A.D.Bell引入的微分算子环D(A；g)。我们证明了：如果A是A的固定子环A‘上的自由模，则D(A；g)同构于D(A’)上m阶矩阵环。根据Grothendieck的一般平坦性定理，对于任意的A，存在A的一个元素c，使得D(A[1/C]；g)同构于D((A[1/c]))上的m乘m方阵环。作为应用，当A是k上的多项式环或Laurent多项式环，g是可对角化的线性自同构时，我们确定了D(A；g)的结构。环R称为(主左)拟Baer环，如果R的每个(主)左理想的左零化子是由一个幂等元生成的。证明了如果R是(左主要)拟…的，G是序么半群的，则么半群环RG又是(左主要)拟Baer环更多的贝尔。当R是(左主要)拟Baer且G是作用在R上的序群时，给出了斜群环R#G是(左主要)拟Baer的一个充要条件。如果环R没有非零幂零元，则称它是约化的。设R是环，g是R的自同构环，我们给出了斜多项式环R[x；g]可约的充要条件，也给出了R[x；g]是Baer环、拟Baer环或主拟Baer环的一些充分条件。设B是Z-Azumaya代数，D是导子，Z‘是Z的元素z，使得D(Z)=0。证明了如果Z/Z‘是指数1的纯不可分扩张，且B满足其他条件，则斜多项式环B[x；D]是Azumaya代数。设p是素数，G是p-群，B是环，f是B的因子集，我们证明了如果交叉积Δ(B，G，f)在B上可分，且p包含在B的Jacobson根内，则Δ(B，G，f)/B是H-可分扩张。我们引入了广义高阶导子的概念，发展了它们与普通高阶导子之间的关系。我们还研究了广义高阶导子的范畴性质。较少