Some results on local Cohomology Modules and Serre subcategories

Abstract

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and let $M$ be a finitely generated $R$-module and $ N $ be an arbitrary $R$-module. Let $n$ be an integer and $\mathcal{S}$ be a Serre subcategory of the category of $R$-modules. We study some properties of local cohomology modules by using the concept of Serre subcategory of the category of $R$-modules. More precisely, we investigate the membership of $\mathrm{H}_\mathfrak{a}^i(M,N)$ and $\mathrm{H}_{Ann_{R}{(M/\mathfrak{a}M)}}^i(N) $ in $\mathcal{S}$ for all $ i<n $ and as the dual case, we find a relation between $\mathrm{H}_\mathfrak{a}^i(M,N) $ and $\mathrm{H}_{Ann_{R}{(N/\mathfrak{a}N)}}^i(M, R) $ for all $ i>n $.