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STUDY OF NOETHERIAN LOCAL RINGS IN COMMUTAIVE ALGEBRA

交换代数中诺特局部环的研究

基本信息

批准号：
10640002
负责人：
NISHIMURA Jun-ichi
金额：
$ 2.11万
依托单位：
Hokkaido University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640002/
关键词：
Noetherian local ring Homological conjectures Frobenius map p-adic representation Bertini Theorem Witt ring big Cohen-Macaulay modules Structure theorem of complete local rings Witt環

项目摘要

Construction of big Cohen-Macaulay modulesHomological conjectures on finitely generated modules over Noetherian local rings are basic and deep problems in commutative algebra. M. Hochster has shown that the existence of a big Cohen-Macaulay module for a given system of parameters of a local ring implies the intersection conjecture and that any local ring which contains a field, or equal characteristic local ring, has such a module.So, to prove the existence of a big Cohen-Macaulay module over unequal characteristic local ring is very important. We have shown that the following study is useful to solve the problem above :1) p-adic representation of elements in an unequal characteristic complete local ring,2) Flenner's Bertini Theorem,3) a lifting of Frobenius map to the Henselization of an unequal characteristic complete local ring.Construction of bad Noetherian local ringsFrom Akizuki's and Nagata's examples it is well-known that some bad examples of Noetherian local rings are meaningful.We have constructed some such local rings, following C. Rotthaus, T. Ogoma and R.C. Heitmann, for example :1) Three dimensional catenary factorial local domain, which is not universally catenary,2) Two dimensional noraml local domain of characteristic 0, which is not analytically reduced,3) Three dimensional local domain of characteristic 0, whose derived normal ring is not Noetherian.

大科恩-麦考莱模的构造诺特局部环上的有限生成模的同调猜想是交换代数中的基本而深刻的问题。 M. Hochster证明，对于给定的局部环参数系统，大Cohen-Macaulay模的存在性暗示了交集猜想，并且任何包含域的局部环，或者相等特征的局部环，都具有这样的模。因此，证明不等特征局部环上大Cohen-Macaulay模的存在性是非常重要的。我们已经表明，以下研究对于解决上述问题是有用的：1）不等特征完全局部环中元素的 p-adic 表示，2）弗伦纳贝尔蒂尼定理，3）将 Frobenius 映射提升到不等特征完全局部环的亨塞尔化。不良诺特局部环的构造从 Akizuki 和 Nagata 的例子中，众所周知，一些不好的例子诺特局部环是有意义的。我们跟随C. Rotthaus、T. Ogoma和R.C. 构造了一些这样的局部环。 Heitmann，例如：1）三维悬链线阶乘局部域，不是普适悬链线，2）特征0的二维正则局部域，没有解析约化，3）特征0的三维局部域，其派生的法环不是诺特环。