Generalized Local Cohomology Modules And Local Cohomology Modules With Respect To A Pair Of Ideals (I, J)

Local cohomology theory given by A.Grothendieck is an important cohomology theory. It is also an an important direction in commutative algebra. Many mathematicians have studied it and have made it well developed. In 1974, Germany mathematician J.Herzog defined generalized local cohomology modules as a generalization of usual local cohomology modules. In a graded ring, we have known that the usual local cohomology modules and generalized local cohomology modules both have the natural graded structures. In 2007, R.Takahashi,Y.Yoshino, T.Yoshizawa defined local cohomology modules with respect to a pair of ideals (I, J) as a generalization of usual local cohomology modules. In this paper, we mainly study generalized local cohomology modules, graded generalized local cohomology modules and local cohomology modules with respect to a pair of ideals (I, J). Taking the special situations in some of our results, we can get some results for usual local cohomology modules.We divide this paper into five chapters. The main content of the paper is the following:In chapter 1, we recall some basic concepts, preliminary results and some famous theorems, which will be probably used in this paper.In chapter 2, we mainly study the weakly laskerian properties, weakly cofiniteness, vanishing properties, attached primes, associated primes and coassociated primes of generalized local cohomology modules. Firstly, we study the relationship between the weakly laskerian properties of H_Iⁱ(M, N) and that of H_Iⁱ(N) and get some equivalent conditions of the weakly laskerian properties of generalized local cohomology modules. Secondly, in a local ring (R,m), we study some properties of a constant quantity q(I,M,N) = sup{i∈N₀|H_Iⁱ(M, N)is not m-weakly cofinite}. After discussing, we find that q(I, M, N) is based on Suppn(N). Thirdly, we study some properties of cd(I,M,N) = sup{i∈N₀|H_Iⁱ(M, N)≠0} and get some vanishing properties of generalized local cohomology modules. Finally, we study the relationship between the attached primes and associated primes of generalized local cohomology modules and discuss the finiteness properties of the coassociated primes of generalized local cohomology modules. In chapter 3, in a graed ring, we study the weakly laskerian properties of graded generalized local cohomology modules and discuss the finiteness properties of associated primes of graded generalized local cohomology modules.In chapter 4, We mainly study the vanishing properties of local cohomology modules with respect to a pair of ideals (I, J) and characterize the coassociated primes of local cohomology modules with respect to a pair of ideals (I, J).In chapter 5, we give other results which including the weakly laskerian properties, weakly cofiniteness of R-modules, local cohomology modules and some Ext-modules, the finiteness properties of associated primes, the connectedness of some set and so on.