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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，我们或多或少达到了同变模同调行为基础研究的目的。在本学年，我们出版了一本英文专著，其中包括该同源研究的技术部分。这部专著将Auslander-Buchweitz关于具有正则模的Cohen-Macaulay环上的Δ-Good逼近理论和Ringel关于拟遗传代数上的Cohen-Macaulay逼近理论统一起来，并从上一学年起进一步丰富了这一定理：如果连通约群G线性作用于多项式代数G上，且S容许好的滤子作为G-模，则不变量环S^G是强F-正则的。由于一个多项式代数在特征零点允许好的滤子的条件总是成立的，并且这个条件是Zariski开的，所以S^G在特征零点是强F-正则型的。由于强F-正则型性质强于奇点的有理性质，因此该定理不包含在Boutot的著名定理中。另一方面，为了研究正特征不变子环的Gorenstein性质，有必要研究正则模的行为。为此，我们证明了关于等变真态射的Grothendieck对偶定理，并构造了描述等变对偶定理所必需的扭转逆伪函子的等变形式。此外，利用它们，我们部分地成功地将Knop关于特征零中不变子环的Gorenstein性质的结果修正为正特征。这些成果在国内外会议上公布，并发表了摘要。此外，我们还得到了一些关于F-有理关于平坦态射的行为的结果，并且已经发表。