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.

通过在上一学年进行的研究，我们或多或少地实现了相同模块同源行为的基本研究的目标。在这个学年，我们发表了一本英语专着，包括该同源研究的技术部分。该专着包括一个理论，该理论将Cohen-MaCaulay近似理论统一了与Auslander-Buchweitz的规范模块的Cohen-Macaulay环相比，Auslander-Buchweitz的规范模块和Δ-GOOOD的理论近似于ringel。越来越多的学年，我们一直在尝试一定的整个学年。代数是线性的，如果S将良好的过滤作为G模块，那么不变的s^g的环是强烈的f-regorumar。由于在特征零中，多项式代数入院良好的过滤始终是正确的，并且条件是Zariski开放的，因此s^g在字符零中具有强f指标类型。由于强大的F型类型属性比奇异性的理性更强，因此该定理不包括在Boutot众所周知的定理中。另一方面，为了研究不变特征的戈伦斯坦特性，有必要研究规范模块的行为。为此，我们就等效的适当形态提供了Grothendieck二元定理，并构建了扭曲的逆伪函数的等效版本，这些版本是指出等效二元定理所必需的。此外，使用它们，我们部分成功地修改了通过诺波旋转到正特征中的零角色中不变级别的结果。这些结果在国内和国际会议上宣布，并发表了摘要。此外，我们在Flat型某些形态方面得到了F理性行为的一些结果，并已发表。