Good filtrations and rings of invariants

良好的过滤和不变量环

基本信息

批准号：
10640017
负责人：
HASHIMORO Mitsuyasu
金额：
$ 1.98万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640017/
关键词：
good filtrations reductive group ring of invariants duality theorem equivariant module Gorenstein property F-rationality strong F-regularity equivariant Gorenstein invariant good filtration duality proper morphism strongly F-re

项目摘要

Through the research done in the last academic year, more or less we achieved the objective on the fundamental research on homological behavior of equivariant modules. In this academic year, we published a monograph in English including the technical part of that homological research. The monograph includes a theory which unifies the Cohen-Macaulay approximation theory over a Cohen-Macaulay ring with canonical modules by Auslander-Buchweitz and the theory of Δ-good approximations over quasi-hereditary algebras by Ringel.Moreover, continued from the last academic year, we were trying to enrich the theorem which asserts that if a connected reductive group G acts on a polynomial algebra S linearly, and if S admits good filtrations as a G-module, then the ring of invariants S^G is strongly F-regular. Since the condition that a polynomial algebra admits good filtrations is always true in characteristic zero, and the condition is Zariski open, S^G is of strongly F-regular type in characteristic zero. As the strong F-regular type property is stronger than rationality of singularities, the theorem is not included in Boutot's well-known theorem. On the other hand, in order to study Gorenstein property of invariant subrings in positive characteristics, it is necessary to investigate the behavior of canonical modules. For this purpose, we proved the Grothendieck duality theorem with respect to equivariant proper morphisms, and constructed the equivariant version of twisted inverse pseudofunctors which are necessary to state the equivariant duality theorem. Moreover, utilizing them, we partly succeeded in modifying the results on Gorenstein property of invariant subrings in characteristic zero by Knop to positive characteristics. These results were announced at domestic and international meetings, and the abstracts were published. Moreover, we got some results on behavior of F-rationality with respect to flat morphisms, and it has been published.

通过上一学年的研究，我们或多或少地达到了对等变模块的同源行为进行基础研究的目标。本学年，我们出版了一本英文专着，其中包括同源研究的技术部分。包括将 Cohen-Macaulay 环上的 Cohen-Macaulay 近似理论与 Auslander-Buchweitz 的规范模统一起来的理论以及准遗传上的 Δ-good 近似理论此外，从上一学年开始，我们试图丰富这个定理，该定理断言，如果一个连通的还原群 G 线性地作用于多项式代数 S，并且如果 S 承认作为 G 模的良好过滤，那么不变量环 S^G 是强 F 正则的，因为多项式代数允许良好过滤的条件在特征零中始终为真，并且该条件为 Zariski。开，S^G在特征零上是强F-正则类型，由于强F-正则类型性质比奇点的有理性更强，因此该定理不包括在著名的Boutot定理中。为了研究正特征中不变子环的 Gorenstein 性质，有必要研究规范模的行为。为此，我们证明了关于等变真值的格罗滕迪克对偶定理。态射，并构造了扭曲逆伪函子的等变版本，这是陈述等变对偶定理所必需的。此外，利用它们，我们部分成功地将 Knop 的关于特征零不变子环的 Gorenstein 性质的结果修改为正特征。在国内和国际会议上发表了论文，并发表了摘要。此外，我们还得到了一些关于F-有理性关于平态射的行为的结果，并且已经发表。