Generalized Local Cohomology Modules

Local cohomology theory is firstly definited by Grothendieck, and later becomes an effective tool in studying commutative algebra and algebric geometry. In 1974, J.Herzog introduced the notation of generalized local cohomology with respect to a pair of modules. In this article, we study the I-weakly cofiniteness and I-weakly co-cofiniteness of generalized local cohomology module HIr(M, N). In addition, we study the weakly co-laskerian property of module ExtR1(R/I, HIr(M, N)).The main content of the paper is the following:In chapter 1, we recall some basic concepts and preliminary results about gener-alized local cohomology, which will be probably used in this paper.In chapter 2, we give the definitions of the W. L module and I-weakly cofinite-ness. Then, we study the Artinianness and I-weakly cofiniteness of generalized local cohomology modules, and give some sufficient conditions. When M is finitely gener-ated, and HIi(N) (i<r) is I-weakly cofiniteness, we discuss the I-weakly cofiniteness of HIr(M,N).In chapter 3, we give the definitions of the W. CL module and I-weakly co-cofiniteness. Then, we study the Artinianness and I-weakly co-cofiniteness of gen-eralized local cohomology modules, and give some sufficient conditions. When M is finitely generated, and HIi(N) (i<r) is I-weakly co-cofiniteness, we discuss the I-weakly co-cofiniteness of HIr(M, N).In chapter 4, we study the weakly co-laskerian property of ExtR1(R/I, HIr(M, N)), and give a good result. When M is finitely generated, and HIi(N)(i<r) is weakly co-laskerian module, we can prove ExtR1(R/I, HIr(M, N)) is a weakly co-laskerian module.