Study of Noetherian Local Rings in Commutative Algebra

Notherian环的反例构造通过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的重要性。然而，它们的施工方法都很复杂，很难得到一个普遍的原则。在这二十年里，由罗特奥斯首创的新的施工方法，由Ogoma和Heitmann发展和简化。在此，我们结合Nagata的思想对这一新工具进行了改进，得到了如下例子：1)三维阶乘局部域，它不是泛连通的；2)特征0的二维正规局部域，它不是解析解分的；3)特征0的三维局部域，它的导生正规环不是Noether环。在理想完备的Nagata环上的Chain条件下，Greco构造了以下令人惊讶的例子：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-进拓扑中是完备的，2)A/I是优的，因此是普适的悬链线。定理1.设(A，m)是一个具有理想I的局部环，假设1)A在I-半拓扑中是完备的，2)A/I是泛链Nagata环。定理2.设A是具有素理想P的Notherian环，假设1)A在P-进拓扑中是完备的，2)A/P是泛链Nagata环。那么，A本身就是普遍的悬链线。

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.