ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

Abstract

This paper is concerned to relationship between the sets of associated primes of the $d$-local cohomology modules and the ordinary local cohomology modules. Let $R$ be a commutative Noetherian local ring, $M$ an $R$-module and $d, t$ two integers. We prove that ${\rm Ass}(H^t_d(M))=\bigcup_{I\in \Phi} {\rm Ass}(H^t_I(M))$ whenever $H^i_d(M)=0$ for all $i< t$ and $\Phi=\{I: I \text{ is an ideal of}\ $R$ \text{ with} \dim R/I\leq d \}$. We give some information about the non-vanishing of the $d$-local cohomology modules. To be more precise, we prove that $H^i_d(R)\neq 0$ if and only if $i=n-d$ whenever $R$ is a Gorenstein ring of dimension $n$. This result leads to an example which shows that ${\rm Ass}(H^{n-d}_d(R))$ is not necessarily a finite set.