STUDY OF NOETHERIAN LOCAL RINGS IN COMMUTAIVE ALGEBRA

大Cohen-Macaulay模的构造Notherian局部环上有限生成模的同伦猜想是交换代数中的基本和深层问题。M.Hochster证明了对于给定的局部环的参数系，存在一个大的Cohen-Macaulay模就蕴含着交猜想，并且任何含有域的局部环或等特征局部环都有这样的模，因此，证明不等特征局部环上存在一个大的Cohen-Macaulay模是非常重要的。我们证明了下列研究对解决上述问题是有用的：1)不等特征完备局部环中元素的p-进表示，2)Flenner的Bertini定理，3)Frobenius映射到不等特征完备局部环的Hensel化的提升。构造坏的Notherian局部环从Akizuki和Nagata的例子众所周知，Notherian局部环的一些坏例子是有意义的。我们构造了一些这样的局部环，例如C.Rotthaus，T.Ogoma和R.C.Heitmann，例如：1)三维悬链线阶乘局部域，它不是普遍的悬链线2)非解析约化的特征0的二维正规局部域，3)导出的正规环不是Noether的三维特征0的局部域。

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.