(J, M)-weakly Regular Sequences And (I, J)-weakly Co-cofinite Modules

Local cohomology is an effective tool in studying commutative algebra and algebric geometry. Many scholars have been absorbed in studying it and have made some effort to develop it. In 2008, R. Takahashi and others introduced the notation of local cohomology with respect to a pair of ideals. In this paper, we study two kinds of regular sequences which influence the properties of modules HI,Ji(M)(i≥0), weakly laskerian properties of modules ExtR(R/I, HI,Jt(M))when i=0,1,2 and (I, J)-weakly co-cofiniteness properties of modules HI,Ji(M)(i≥0). We divide this paper into three chapters.The main content of the paper is the following:In chapter 1, we recall some basic concepts, preliminary results and some famous theorems about local cohomology with respect to a pair of ideals and coassociated primes, which will be probably used in this paper.In chapter 2, we define the definition of (J, M)-regular elements, (J, M)-regular sequences, (J, M)-weakly regular elements, and (J,M)-weakly regular sequences. In view of those,we discuss some fundamental and developed properties. Mainly we have:if HI,Ji(M)=0,(?)i1)? In the end,we show the important conclusion that if HI,Ji(M)=0, (?)i1).In chapter 3, we define the modules which were called (I, J)-weakly co-cofinite modules in chapter 3. In this chapter, we study some properties of (I, J)-weakly co-cofinite modules. Mainly we have:we get some sufficient conditions of the modules ExtRi(R/I, HI,Jt(M)) which are (I, J)-weakly co-Laskerian modules? when i=0,1,2. In the end, we show a satified result of when the local cohomology modules with respect to a pair of ideals is (I, J)-weakly co-cofinite?...