模情形下有限群的不变式环的Cohen-Macaulay性质、余不变式环及Noether数的研究

在1900年国际数学家大会上，Hilbert提出的第14个问题是不变量理论的核心问题：任意群作用在任意域上的多项式代数得到的不变式环总是有限生成的吗？尽管Nagata给出了反例，但Noether在1915年证明了有限群的不变式环总是有限生成的，这奠定了有限群不变量理论的基础。本项目致力于研究模情形下(基域的特征整除群的阶数)有限群的不变式环的Cohen-Macaulay性质、余不变式环和Noether数。具体的研究内容为：有限群的余不变式环的结构；由2-反射生成的p-群的不变式环的Cohen-Macaulay性质及Kemper猜想；三类有限群的有限维表示的Noether数。这些问题不仅是重要的，而且具有相当的难度。但我们将利用有限群的上同调、SAGBI基等工具解决上述问题。本项目的完成有助于加深对模不变量理论的认识，并促进对其它问题(比如不变式环的Gorenstein性质等)的研究。

At the International Congress of Mathematicians in 1900, the 14-th problem Hilbert proposed was a central problem of the invariant theory: is the ring of invariants obtained by an arbitrary group action on polynomial algebras over arbitrary fields always finitely generated? Although, Nagata gave a counter-example, Noether proved that the invariant rings of finite groups are finitely generated, which lays the foundation of the invariant theory of finite groups. This project is dedicated to research the Cohen-Macaulay property of the invariant rings, the coinvariant rings and Noether numbers of a finite group in the modular case(the characteristic of the field divides the order of a finite group). The specific research contents are as follows: the structures of the coinvariant rings of finite groups; the Cohen-Macaulay property of the invariant rings of p-groups which are generated by bireflections and Kemper's conjecture; the Noether numbers of non-trivial finite dimensional representations of three kinds of finite groups. Those problems are not only important, but very difficult. However,we will use tools such as group cohomology of finite groups and the SAGBI basis to solve the above problems. The completion of this project will help to deepen the understanding of the modular invariant theory and promote the study of more general problems(such as the Gorenstein property of invariant rings and so on) in the modular case.