On the associated primes of the generalized d-local cohomology modules

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized d-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that R is a commutative Noetherian local ring, M and N  are  finitely generated  R-modules and d, t are two integers. We prove that \Ass H^t_d(M,N)=\bigcup_{I\in \Phi} \Ass H^t_I(M,N) whenever H^i_d(M,N)=۰ for all  i< t and \Phi=\{I: I  \text{ is an ideal of}\  R \text{ with}\ \dim R/I\leq d \}. In the second part of the paper, we give some information about  the non-vanishing of the generalized d-local cohomology modules. To be more precise, we prove that H^i_d(M,R)\neq ۰ if and only if i=n-d whenever  R is a Gorenstein ring of dimension n and pd_R(M)<\infty. This result leads to an example which shows that \Ass H^{n-d}_d(M,R) is not necessarily a finite set.

کلیدواژه ها:

Gorenstein Ring ، Associted prime ideals ، d-local cohomology modules