Weakly Co-cofinite Modules, Gorenstein Injective Modules And Local Cohomology

Let(R,m)be a commutative Noetherian local ring with unit element and M an Rmodule. For each i≥0,we denoted the ith local cohomology module of M with respect to a by H_a~i(M).For more details of this concept,please refer to[15].In[20],the author introduced the notation of weakly co-Laskerian module,following the notation of a-cofinite and a-weakly cofinite we define the modules which called a-weakly co-cofinite module in chapter 2.Then we continue to study some properties of a-weaky co-cofinite modules in chapter 2 and raise the following question:When the module Ext_R~i(R/a,H_a~i(M)) is a-weakly co-cofinite module? We discuss the question in the case i= 0,1,2.In the end, we show a satisfied result of when the local cohomology module is a-weakly co-cofinite?In chapter 3,we study the relations between Gorenstein injective module(GI for short) and local cohomology.We give two vanishing theorems of local cohomology and generalized local cohomology of modules with finite Gorenstein injective dimensions.If M is GI(SGI), we also solve the question whenΓ_a(M)is GI(SGI).