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

Abstract

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 $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)$ whenever $H^i_d(M,N)=0$ for all  $i< t$ and $Phi={I: I  text{ is an ideal of}  R text{ with} dim R/Ileq d }$. 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 $H^i_d(M,R)neq 0$ 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.