ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

Abstract

Let  $R$ be a commutative Noetherian ring,  $I, J$ two proper ideals of

$R$ and let $M$ be a non-zero finitely generated  $R$-module with $c={\rm cd}(I,J,M)$.

In this paper, we first  introduce $T_R(I,J,M)$ as the largest submodule of $M$

with the property that ${\rm cd}(I,J,T_R(I,J,M))<c$ and we describe it in terms of the reduced primary

decomposition of zero submodule of $M$. It is shown that

 ${\rm Ann}_R(H_{I,J}^d(M))={\rm Ann}_R(M/{T_R(I,J,M)})$ and ${\rm Ann}_R(H_{I}^d(M))={\rm Ann}_R(H_{I,J}^d(M))$,

whenever $R$ is a  local ring, $M$ has dimension $d$ with $H_{I,J}^d(M)\\\neq0$ and

$J^tM\subseteq T_R(I,M)$ for some positive integer $t$.

Keywords

• Attached prime ideal,top local cohomology module

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