Local cohomology modules and Cousin complexes

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- ۱}(\operatorname{C}_R(\mathcal{F}, X))) and \operatorname{H}^{s'- ۱}(\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.

Keywords:

Cousin complexes , local cohomology modules , Serre subcategories