On The Properties And Applications Of The Uniform Local Cohomological Annihilator

This paper mainly studys the properties and applications of the uniform local coho-mological annihilator. It is well-known that if a noetherian ring R has a uniform local cohomological annihilator, then R[X] also has a uniform local cohomological annihilator. One of the main results of the paper is to prove that the converse of the above conclusion is also true, which means if R[X] has a uniform local cohomological annihilator, then R will also have a uniform local cohomological annihilator. It shows that the property of having a uniform local cohomological annihilator is a universal property. The another main result of the paper is concerning with monomial conjecture. We will establishes a close connection between the uniform local cohomological annihilator and the monomial conjecture. By using this connection, we are able to give a simple proof of the monomial conjecture in the case when the local ring contains a finite field.This article is mainly divided into the following six parts.The first chapter is the introduction, in which we will give the background and some notion of the paper.In the second chapter, we recall a series of concepts and properties which will be used in the paper.In the third part, we will recall the notion of the uniform local cohomological annihi-lator, and list some important theorems which will be used in the rest of the paper.In the forth chapter, we will prove that the property of a noetherian ring R having a uniform local cohomological annihilator is a universal property, and this is one of the main results of the paper.In the fifth chapter, we present a deep relationship between the uniform local coho-mological annihilator and the monomial conjecture. In the final chapter, we will use the results developed in the last chapter to give a simple proof of the monomial conjecture in the case of the local ring contains a finite field.