Commutative ring theory and singularity theory

1.乘子理想；J.Lipman和K.Watanabe证明了二维正则局部环上的每个积分闭理想都是乘子理想；N.Hara和K.Yoshida定义了特征p&gt；0中“紧闭包”的一个推广，并利用这个概念成功地用纯代数方法(用交换环论)计算了乘子理想；S.Takagi和K.Watanabe建立了“F-纯门限”的概念，它对应于特征0中用于代数几何的lc(=对数典范)门限的概念.这一概念在奇点理论和交换环理论中都有许多有趣的特征。2.Hilbert-ukz重数是定义在具有正特征的环上的一种重数。Watanabe和Yoshida以前证明了环是正则的当且仅当环的HK重数为1。这一次我们确定了2维和3维非正则环中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.