ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

Year 2019, , 55 - 63, 08.01.2019

Z. Rahimi-molaei Sh. Payrovi S. Babaei

https://doi.org/10.24330/ieja.504110

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.

Keywords

Associated primes, vanishing theorem, d-local cohomology module

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