STUDY OF NOETHERIAN LOCAL RINGS IN COMMUTAIVE ALGEBRA

大Cohen-Macaulay模的构造noether局部环上有限生成模的同调猜想是交换代数中基本而深刻的问题。M. Hochster证明了局部环的给定参数系的大Cohen-Macaulay模的存在性蕴涵了交猜想，并且证明了任何包含域的局部环或等特征局部环都有这样的模。因此，证明一个大Cohen-Macaulay模在不等特征局部环上的存在性是非常重要的。我们证明了以下研究对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进表示，2)Flenner的Bertini定理，3)将Frobenius映射提升到不等特征完全局部环的Henselization。坏Noetherian局部环的构造从Akizuki和Nagata的例子中，我们知道一些Noetherian局部环的坏例子是有意义的。我们在C. Rotthaus， T. Ogoma和R.C. Heitmann的基础上构造了一些这样的局部环，例如：1)三维链链线阶乘局部域，它不是普遍链链线；2)特征0的二维norml局部域，它不是解析约简的；3)特征0的三维局部域，它的导出正规环不是noether的。

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.