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Study of Noetherian Local Rings in Commutative Algebra

交换代数中诺特局部环的研究

基本信息

批准号：
03640045
负责人：
NISHIMURA Jun-ichi
金额：
$ 1.15万
依托单位：
HOKKAIDO UNIVERSITY OF EDUCATION (1993)Kyoto University (1991-1992)
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1993
项目状态：
已结题

项目摘要

Construction of Counter-Examples of Noetherian RingsThrough famous examples due to Akizuki and to Nagata, in commutative Noetherian ring theory, it is well-known that constructing counter-examples is no less important than showing positive results. However, thier methods of construction were complicated and hard to get a general principle.For these twenty years, the new construction method, originated by Rotthaus, have been and developed and simplified by Ogoma and by Heitmann. This new construction enables us not only to reconstruct easily known examples but to obtain new unknown examples, which give answers to a number of open problems.Here, we improve this new tool, combining ideas of Nagata, and get the following examples :1)3-dimensional factorial local domain which is not universally catenaty.2)2-dimensional normal local domain of characteristic 0 which is not analytically unramified.3)3-dimensional local domain of characteristic 0 whose derived normal ring is not NOetherian.Chain Conditions on Ideal-adically Complete Nagata RingsGreco has constructed the following surprising example :Example. There exists a semi-local domain (A,m_1, m_2) with an ideal I=P_1 * P_2 (= the intersection of two prime ideals) such that 1) A is complete in I-adic topology, and 2) A/I is excellent, hence universally catenary. But A itself is not universally catenary.On the other hand, we get the following :Theorem 1. Let (A,m) be a local ring with an ideal I.Suppose that 1) A is complete in I-adictopology, and 2) A/I is a universally catenary Nagata ring. Then, A itself is universally catenary.Theorem 2. Let A be a Noetherian domain with a prime ideal P.Suppose that 1) A is complete in P-adic topology, and 2) A/P is a universally catenary Nagata ring. Then, A itself is universally catenary.

构造Noether环的反例通过Akizuki和Nagata的著名例子，在交换Noether环理论中，众所周知，构造反例的重要性不亚于证明正结果。但其施工方法复杂，难以得到普遍的原则，近20年来，这种新的施工方法由Rotthaus首创，经过Ogoma和Heitmann的发展和简化。这种新的构造不仅使我们能够重建容易知道的例子，而且还可以获得新的未知例子，这些例子可以回答一些开放的问题。在这里，我们改进了这个新工具，结合Nagata的思想，并得到以下例子：1）三维阶乘局部域不是泛链的，2）特征为0的二维正规局部域不是解析不可分歧的，3）特征为0的三维局部域其导出正规环不是诺特环的. Greco构造了以下令人惊讶的例子：存在一个半局部整环（A，m_1，m_2），其理想I=P_1 * P_2（=两个素理想的交）使得1）A在I-adic拓扑中是完备的，2）A/I是优的，因而是泛悬链线的.但A本身并不是泛悬链线。另一方面，我们得到如下结果：定理1。设（A，m）是具有理想I的局部环，设1）A是I-自根拓扑完备的，2）A/I是泛悬链线Nagata环.则A本身是泛悬链线。定理2。设A是具有素理想P的Noether整环，1）A是P-adic拓扑完备的，2）A/P是泛悬链线Nagata环。那么，A本身就是泛悬链线。