Castelnuovo-Mumford Regularity Of The Ideal Product Of Irreducible Monomials

Let k be a field,and let S = k[x₁,…,x_n]be the polynomial ring in n variables over k.An ideal I of S is called irreducible monomial ideal,if I is generated by the power of the indeterminate variables of the S,such as I=?x₁²,x₃²,x₅⁶?.The irreducible monomial ideal is a special kind of complete intersection monomial ideal.Through the method of induction for the number of minimal sets of generators of the monomial ideal and giving proof of some related lemmas,it is proved in this thesis that the Castelnuovo-Mumford regularity of the product of four irreducible monomial ideals I,J,K and L meets the inequation reg?IJKL?<reg?I?+reg?J?+reg?K?+reg?L?,where reg?I?represents the Castelnuovo-Mumford regularity of I.The main contents in the thesis are as follows:The first chapter briefly introduces the research background and developmen-t status of Castelnuovo-Mumford regularity of irreducible monomial ideals.The chapter also presents the problem to study in the thesis.The second chapter is the preliminary knowledge,which enumerates the main tools used in this research.These five lemmas are the support of this study,especially lemma 2.1.2 and lemma 2.1.3 are repeatedly used in the article;The lemma 2.2.1 and lemma 2.2.2 are the prerequisites for this study.In chapter 3,there are nine lemmas based on the method of induction.These 9 lemmas are found in the process of proof.Because of the theoretical needs of this study,these 9 lemmas are proved firstly.The fourth chapter is the key of this paper and all the previous studies have laid the foundation for this chapter.In this chapter,it is proved that reg?IJKL?<reg?I?+reg?J?+reg?K?+ reg?L?by using the method of the induction?the above mentioned lemma and preparatory knowledge.The I,J,K and L are irreducible monomial ideals in a polynomial ring over a field k.