Gorenstein dimension and Gorenstein algebras

Gorenstein 维数和 Gorenstein 代数

• 批准号: 17540021
• 负责人: NISHIDA Kenji
• 金额: $ 2.11万
• 依托单位: Shinshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540021/
• 关键词: Gorenstein dimension; Gorenstein ring; filtered ring; graded ring; grade; Auslander condition; holonomic module; totally reflexive module; 双対化加群; 半双対化加群; 入射次元; コ-エン-マコーレー加群

We consider noncommutative filtered rings and filtered modules over them. Our main objective is to calculate homological invariants and some equalities or inequalities of homological invariants, like Gorenstein dimension, grade. When this is the case, we see that associated graded rings and modules play important role to handle such homological invariants. We get that Gorenstein dimension of an associated graded module grM is greater than or equal to that of a filtered module M over a filtered ring. We also get that, under general condition as before, grade of an associated graded module coincides with that of a filtered module. This condition is described by module-wise condition. Hence we can study grade of modules over not necessarily Gorenstein filtered rings and see that there is a duality given by Ext-group between left and right Cohen-Macaulay modules over such filtered ring. The properties of Gorenstein filtered rings are also studied under the point of view of generalization of properties of Auslander regular ring. We get, for example, that a module is pure if and only if it is geometrically pure and has no isolated associated primes.

我们考虑非交换滤环和滤模。我们的主要目标是计算同调不变量以及同调不变量的一些等式或不等式，如Gorenstein维数、阶数等。当这种情况下，我们看到，相关的分次环和模起着重要的作用，处理这样的同调不变量。我们得到了滤环上相伴分次模grM的Gorenstein维数大于等于滤环上滤模M的Gorenstein维数。我们还得到，在一般条件下，一个相联分次模的阶数与一个滤子模的阶数一致。这个条件用模条件来描述。因此，我们可以研究不一定Gorenstein滤环上模的阶，并发现在这种滤环上的左、右Cohen-Macaulay模之间存在一个由Ext-群给出的对偶。从Auslander正则环的性质推广的角度研究了Gorenstein滤环的性质。例如，我们得到一个模是纯的当且仅当它是几何纯的并且没有孤立的相伴素数。

项目成果

Homological properties over filtered rings

过滤环上的同源性质

• 发表时间: 2007
• 作者: H. Miyahara;K. Nishida;H. Miyahara(K. Nishida)
• 通讯作者: H. Miyahara(K. Nishida)

A generalization of the Auslander formula

Auslander 公式的推广

• 发表时间: 2005
• 期刊: Fields Institute Communications Vol.45
• 作者: M.Hoshino;K.Nishida
• 通讯作者: K.Nishida

Cohen-Macaulay isolated singularities with a dualizing module

使用对偶模块 Cohen-Macaulay 隔离奇点

• 发表时间: 2006
• 期刊: Algebras and Representation Theory 9
• 作者: H. Miyahara;K. Nishida;K. Nishida;R. Takahashi;K. Nishida
• 通讯作者: K. Nishida

A characterization of modules locally of finite injective dimension

有限单射维数局部模的表征

• 发表时间: 2007
• 期刊: Proceedings of the American Mathematical Society 135,no.11
• 作者: Shiro Goto;Naoyuki Matsuoka and Ryo Takahashi;Ryo Takahashi;Shunsuke Takagi and Ryo Taka hashi;Ryo Takahashi;Ryo Takahashi
• 通讯作者: Ryo Takahashi

Finite groups with multiplicity-freepermutation characters

具有无重数排列特征的有限群

• 期刊: J.Alg.Its Appl. (in press)
• 作者: M.Fuma;Y.Ninomiya
• 通讯作者: Y.Ninomiya