Hilbert Coefficients Of Hilbert Filtrations And Generalized Local Cohomology Modules

In the past two decades,many mathematicians such as S.Huckaba,S.Goto,C.Huneke and S.Zarzuela have extensively studied blowup algebras of I(where(R,m) is a commutative and Noetherian local ring with infinite residue field,I is an ideal of R),especially their depth properties,and they have given many important results,these results are also used to examine diverse properties of the ideal I.In chapter 2,we give a generalization of the above ideal I,we mainly discuss the interplay between the depths of the associated module G(F,M) of F with respect to M and the fiber cone F_K(F,M) of F with respect to M and K(where M is a finitely generated R-module, F= {I_n}_n≥0 is a Hilbert filtration with respect to M,K is an m-primary ideal of R such that I_n+1(?) KI_n for all n≥0) and the Hilbert coefficients of F.We mainly get the following results in chapter 2:Firstly,we describe the Hilbert coefficients of F when G(F,M) and F_K(F,M) have almost maximal depth.Secondly,we give an upper bound on the Hilbert series of F with respect to M and K.When Hilbert series get the bound,we discuss the depths of F_K(F,M) and Sally module S_J(F,M)(where J is a minimal reduction of F with respect to M),the Hilbert coefficients of F and S_J(F,M).Finally,we discuss some results on mixed multiplicities.Local cohomology is a useful tool in several branches of commutative algebra and algebraic geometry.An important problem in commutative algebra is to determine when the set of associated primes of the ith local cohomology modules H_Iⁱ(M) is finite(where integer i≥0). This question has been studied by many reserchers.A generalization of local cohomology functors has been given by J.Herzog[32]in 1974,this notion is a generalization of the usual local cohomology functor.In this paper,we mainly study the Artinianness,I-cofiniteness, weakly Laskerianness of generalized local cohomology modules and the finiteness of the set of associated primes.The chapter 3 of this paper is following:Firstly,we study the Artinianness and I-cofiniteness of generalized local cohomology modules,and get some sufficient conditions on the I-cofiniteness of H_I^r(M,N)(where M and N are two R-modules,r is a non-negative integer).Then we discuss the weakly Laskerianness of generalized local cohomology modules,and we get the sufficient conditions on the weakly Laskerianness of Hom_R(R/I,H_I^r(M,N)) and Ext_R¹(R/I,H_I^r(M,N)) and the results on the finiteness of corresponding sets of associated primes.Finally,by the definitions and properties of k-regular sequence and k-depth,we study the finiteness of the set of associated primes of generalized local cohomology modules on some conditions.The set of attached primes of generalized local cohomology modules will be discussed in this chapter.