Good filtrations and rings of invariants

通过上一学年的研究，我们或多或少达到了对等变模的同调行为进行基础研究的目的。在本学年，我们出版了一本英文专著，包括该同源性研究的技术部分。本专著将Auslander-Buchweitz提出的具有正则模的Cohen-Macaulay环上的Cohen-Macaulay近似理论与Ringel提出的拟遗传代数上的Δ-good近似理论统一起来。此外，从上一学年开始，我们试图充实一个定理，即如果一个连通约化群G线性作用于一个多项式代数S，如果S作为一个G模允许良好的过滤，那么不变量环S^G是强f正则的。由于多项式代数允许良好滤波的条件在特征零上总是成立的，且该条件是Zariski开的，因此S^G在特征零上是强f正则型。由于强f正则型性质比奇点的合理性更强，所以该定理没有包含在Boutot的著名定理中。另一方面，为了研究正特征中不变量子函数的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固有态射的Grothendieck对偶定理，并构造了说明等变对偶定理所必需的扭曲逆伪函子的等变版本。此外，利用它们，我们部分成功地将Knop关于特征0中不变量子向量的Gorenstein性质的结果修正为正特征。这些结果在国内和国际会议上公布，并发表了摘要。此外，我们还得到了关于平面态射的f -理性行为的一些结果，并已发表。

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.