Local Cohomology Modules And Generalized Local Cohomology 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 1974,Germany mathematician J.Herzog defined generalized local cohomology modules as a generalization of usual local cohomology modules.In this paper,we mainly study the finiteness,Artinianness and tameness properties of(generalized) local cohomology modules and get some meaningful results.We divide this paper into four 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 and generalized local cohomology modules,which will be probably used in this paper.In chapter 2,we study the Artinianness of(generalized) local cohomology modules. Firstly,we study the relationship between the Artinianness of H_I^T(R/p)((?)p∈Supp(N))and that of H_Iⁱ(N)((?)i≥r),determine the last non-Artinian local cohomology module,and generalize it to the generalized local cohomology case.Secondly,we discuss the relationship between the Artinianness of H_Iⁱ(M,N) and that of H_Iⁱ(N).Finally,we determine the first non-Artinian generalized local cohomology module by the length of a maximal filter regular sequence.In chapter 3,we mainly study the finiteness and cofiniteness properties of generalized local cohomology modules.In[26],Hartshorne posed the following question:when is H_Iⁱ(N) cofinite? In the generalized local cohomology case,Yassemi also posed the similar question.In this direction,we give some results.And we also give some finiteness and vanishing results for generalized local cohomology modules under some conditions.In chapter 4,we mainly study the tameness of graded(generalized) local cohomology modules.For the local cohomology module H_R+ⁱ(N),we have the well-known results:(â…°) for all i and all n,R₀-module H_R+ⁱ(N)_n is finitely generated;(â…±) for all i,end(H_R+ⁱ(N))＜∞. The following question is asked naturally:generally,for a graded ideals I contained in R₊,when does H_Iⁱ(N) have the similar results? We study this question and give a result about the tameness of H_Iⁱ(N).At the end,we give some results about the tameness of generalized local cohomology modules which generalize some known results in[54]and [18].