THE GLUING OF MAXIMAL IDEALS- SPECTRUM OF A NOETHERIAN RING-GOING UP AND GOING DOWN IN POLYNOMIAL RINGS
THE GLUING OF MAXIMAL IDEALS- SPECTRUM OF A NOETHERIAN RING-GOING UP AND GOING DOWN IN POLYNOMIAL RINGS
复制标题
最大理想的粘合-诺特环的谱-多项式环中的上升和下降
DOI：
10.1090/s0002-9947-1980-0574801-5
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发表时间：
1980
影响因子：
1.3
通讯作者：
Y. Lequain
中科院分区：
文献类型：
--
作者：
Ada Maria de Souza Doering;Y. Lequain
如果山，。. .,是环R的极大理想，且R具有同构的剩余域，则它们可以是“胶合”的，即R的子环D在D上是整的且MxnD =. . = M，n D可以被构造。我们使用这个粘合过程来证明以下结果：给定任何有限序集®，存在约化Noether环B和嵌入i//：··规范B，使得i//建立最大（分别为最小）元素%和最大（分别为极小）素理想B，使得给定的任何元素s '，s”，在uV（1/3“）和>K/3”之间存在长r的素理想的饱和链当且仅当i在s“和s”之间存在长r的饱和链。我们还利用胶合过程构造了一个具有商域L的Noether整环A和一个位于A和L之间的Noether整环B，使得：A“^ B具有上升和下降性质，A（X）c* B（X）是单支的，A（X）B（X）既不具有上升也不具有下降性质. 1.导言和定义。在（11）中，Nagata构造了他著名的局部区域D不满足高度公式的例子。构造的基本思想是考虑域R的两个不同高度的极大理想AT和M2，并在构造满足以下两个性质的R的子环D的意义上将它们“粘合”：R是D上的整数且Mx n D = M2 n D。永田指出，如果7？包含一个域K，它是RM和RM的代表域，那么Mx和M2确实可以通过取D = K +（Mx n M2）来粘合;他还指出，如果R是诺特的，则D是诺特的。主要的困难，永田遇到的是拿出一个诺特域R有两个最大的理想Mx和M2的不同高度，他的胶合过程可以执行。很明显，这样一个环R不可能是一个有限Tv-代数的幂元环，它以一种非常巧妙的方式与形式幂级数一起工作，Nagata构造了一个。
If Mt,. . ., M, are maximal ideals of a ring R that have isomorphic residue fields, then they can be "glued" in the sense that a subring D of R with R is integral over D and Mx n D = .. . = M, n D can be constructed. We use this gluing process to prove the following result: Given any finite ordered set ®, there exists a reduced Noetherian ring B and an embedding i//: ® -» Spec B such that i// establishes a bijection between the maximal (respectively minimal) elements of % and the maximal (respectively minimal) prime ideals of B and such that given any elements s', s" of ®, there exists a saturated chain of prime ideals of length r between uV(/3') and >K/3") if and only i there exists a saturated chain of length r between s' and s". We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that: A "^ B possesses the Going Up and the Going Down properties, A(X)c* B(X) is unibranched and A(X) «-» B(X) possesses neither the Going Up nor the Going Down properties. 1. Introduction and definitions. In (11), Nagata constructed his famous example of a local domain D that does not satisfy the altitude formula. The basic idea of the construction is to consider two maximal ideals AT, and M2 of different height of a domain R and to "glue" them in the sense of constructing a subring D of R that satisfies the following two properties: R is integral over D and Mx n D = M2 n D. Nagata noted that if 7? contains a field K that is a field of representatives for both RM and RM, then Mx and M2 can indeed be glued by taking D = K + (Mx n M2); he also noted that D is Noetherian if R is Noetherian. The main difficulty that Nagata encountered was to come up with a Noetherian domain R having two maximal ideals Mx and M2 of different height on which his gluing process could be performed. It is clear that such a ring R cannot be a ring of quotients of a finite Tv-algebra, and it is working in a very clever way with formal power series that Nagata constructed one.