# Local cohomology modules and Cousin complexes

Document Type : Research Paper

Authors

Department of Mathematics, Payame Noor University (PNU), P.O.BOX, 19395-4697, Tehran, Iran

10.29252/as.2019.1607

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.

Keywords

#### References

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