ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

Authors : Z RAHİMİMOLAEİ, Sh PAYROVİ, S BABAEİ

Pages : 55-63

Doi:10.24330/ieja.504110

View : 15 | Download : 8

Publication Date : 2019-01-08

Article Type : Research Paper

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  ${\rm Ass}insert ignore into journalissuearticles values(H^t_dinsert ignore into journalissuearticles values(M););=\bigcup_{I\in \Phi} {\rm Ass}insert ignore into journalissuearticles values(H^t_Iinsert ignore into journalissuearticles values(M););$ whenever $H^i_dinsert ignore into journalissuearticles values(M);=0$ 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 $H^i_dinsert ignore into journalissuearticles values(R);\neq 0$ 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 ${\rm Ass}insert ignore into journalissuearticles values(H^{n-d}_dinsert ignore into journalissuearticles values(R););$ is not necessarily a finite set.
Keywords : Associated primes, vanishing theorem, d local cohomology module