Singurality Theory and Frobenius Morphism

特征p > 0的Frobenius自同态是交换环理论、奇点理论或特征为0的域上代数几何中通过模p约化的一个非常有力的工具。具体而言，我们展示了以下内容：1。F-阈值;定义了特征p>0的Noether环中A的两个理想对（I，J）的F-阈值。这个概念最初是用来描述正则局部环中的乘子理想和跳跃数的。但在我们的研究中，发现这个概念与紧闭包和积分闭包密切相关，并且我们对参数理想的F-阈值和重数有一个很好的猜想.多重分次环、有理奇异性和F-有理环;多重分次环及其对角代数的概念是一个非常有趣的对象，在制作许多有趣的例子时非常有用。在与A Singh和E.佐藤角Kurano和K. Watanabe给出了一个新的例子，离散除子群的局部上同调模表现出非常有趣的特征。给出了多重分次超曲面的对角子代数是f-有理或F-正则的一个度判据.全自反模：在全自反模理论中，非平凡全自反模的例子很少。Watanabe和R. Takahashi利用亏格大于1的曲线几何构造了一类非平凡的全自反模。这是第一个情况下，代数几何是在这个理论中使用。

Frobenius endomorphism hi characteristic p > 0 is a very powerful tool in commutative ring theory as well as singularity theory or algebraic geometry iover a field of characteristic 0 via reduction mod p.We applied the Frobenius endomorphism to various problems in commutative algebra and singularity theory. Inparticular, we showed the followings ;1. F-thresholds ; F-threshold is defined in a Noetherian ring of characteristic p>0 to a pair (I,J) of two ideals of A.This notion was originally introduced to describe multiplier ideals and jumping numbers in a regular local ring. But in our research, it turned out that this notion is closely related to tight closures and integral closures and also we have a nice conjecture concerning F-threshold and multiplicity of a parameter ideal.2. Multi-graded rings, rational singularity and F-rational rings ;The notion of multi-graded rings and their diagonal algebras is a very interesting object and very useful in making many interesting examples. In a joint paper with A Singh and E. Sato, K. Kurano and K. Watanabe made a new example with discrete divisor class group whose local cohomology modules shows very interesting feature. Also we showed a criterion for diagonal subalgebras of multi-graded hypersurfaces to be f-rational or F-regular in terms of the degree.3. Totally reflexive modules ;In the theory of totally reflexive modules, examples of non-trivial totally reflexive modules are very few. Watanabe and R. Takahashi constructed a family of non-trivial totally reflexive modules using geometry of curves of genus greater than 1. This is the first case that algebraic geometry is used in this theory.