Local cohomology modules and Cousin complexes

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $mathfrak{a}$ an ideal of $R$, $X$ an arbitrary $R$--module, $mathcal{F}$ a filtration of $operatorname{Spec}(R)$ which admits $X$, and $s, s', t, t'$ non-negative integers such that $s+ t= s'+ t'$. In this paper, we study the membership of $R$--modules $operatorname{H}^{s}_mathfrak{a}(operatorname{H}^{t- 1}(operatorname{C}_R(mathcal{F}, X)))$ and $operatorname{H}^{s'- 1}(operatorname{H}^{t'}_mathfrak{a}(operatorname{C}_R(mathcal{F}, X)))$ in Serre subcategories of the category of $R$--modules and find some sufficient conditions which ensure the existence of an isomorphism between them, where $operatorname{C}_R(mathcal{F},X)$ is the Cousin complex for $X$ with respect to $mathcal{F}$. As applications, we give some new facts and represent some older facts about the local cohomology modules and the Cousin complexes.