Study of Noetherian Local Rings in Commutative Algebra

Noetherian环反例的构造通过Akizuki和Nagata的著名例子，在交换Noetherian环理论中，构造反例的重要性不亚于给出正结果。然而，他们的构造方法复杂，难以得到一个普遍的原则。这二十年来，由罗特豪斯首创的新施工方法，经过Ogoma和Heitmann的发展和简化。这种新的结构使我们不仅能够重构容易已知的例子，而且能够获得新的未知的例子，从而为许多开放的问题提供答案。在此，我们结合Nagata的思想，对这个新工具进行了改进，得到了以下例子：1)三维阶乘局部域，它不是普遍存在的。2)特征为0的二维法向局部域，其特征为非解析无分支。3)特征0的三维局部区域，其导出的正规环不是noether的。理想完全永田环上的链条件sgreco构造了以下令人惊讶的例子：存在一个理想I=P_1 * P_2（=两个素数理想的交集）的半局部域（a, m1, m_2），使得1)a在I进拓扑上是完全的，2)a /I是优秀的，因此是普遍链线。但是A本身并不是普遍的链线。另一方面，我们得到如下结论：定理1。设（A,m）是一个具有理想I的局部环，设1)A在I-adictopology上是完全的，2)A/I是一个普适链线Nagata环。那么，A本身是普遍链线。定理2。设A是一个具有素数理想P的noether域，设A在P进拓扑上是完全的，且A/P是一个普遍链线永田环。那么，A本身是普遍链线。

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.