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

Document Type : Research Paper

Authors

1 Department of Mathematics, Imam Khomeini International University, Qazvin, Iran

2 Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.

3 Department of mathematics, Imam Khomeini International University, Qazvin, Iran

10.29252/as.2020.1615

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

Keywords


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