Commutative ring theory and singularity theory

结果主要涉及以下3个方面：1.乘子理想; J.Lipman和K.Watanabe证明了2维正则局部环中的每一个整闭理想都是乘子理想，N.Hara和K.Yoshida定义了特征p>0的“紧闭包”的推广，并利用这一概念，他们成功地计算乘子理想的纯代数S.Takagi和K.Watanabe建立了“F-纯阈值”的概念，其对应于在代数几何中使用的特征0中的LC（=对数正则）阈值的概念。这个概念在奇点理论和交换环理论中都有许多有趣的特征。2.希尔伯特-昆兹重数：希尔伯特-昆兹重数是定义在正特征环上的一种重数。Watanabe和Yoshida证明了环是正则的当且仅当环的HK重数是1。这一次我们确定了在2维和3维的非正则环中HK重数最小的环。

The results are mainly concerning the followings 3 themes1.Multiplier ideals ;J.Lipman and K.Watanabe proved that every integrally closed ideal in 2 dimensional regular local rings is a multiplier ideal.N.Hara and K.Yoshida defined a generalization of "tight closures" in characteristic p>0 and by using that concept, they Succeeded to calculate multiplier ideals by purely algebraic (by commutative ring theory) method.S.Takagi and K.Watanabe established the notion of "F-pure thresholds", which corresponds to the notion of lc(=log canonical) threshold in characteristic 0, used in algebraic geometry. This concept has many interesting features in both singularity theory and commutative ring theory.2.Hilbert-Kunz multiplicity ;Hilbert-Kunz multiplicity is a kind of multiplicity defined for rings of positive characteristics. Watanabe and Yoshida proved before that a ring is regular if and only if the HK multiplicity of the ring is 1. This time we determined the rings whose HK multiplicity is smallest among non-regular rings in dimension 2 and 3.