Serre subcategory, generalized local homology and generalized local cohomology modules

This paper deals with generalized local homology and generalized local cohomology modules belong to a Serre category of the category of $R$-modules under some conditions. For an ideal $I$ of $R$, the concept of the condition $C^I$ on a Serre category which is dual to the condition $C_I$ of Melkersson is defined. As a main result, it is shown that for a finitely generated $R$-module $M$ with $pd(M) <\infty$ and a minimax $R$-module $N$ of any Serre category $\mathcal{S}$ satisfying the condition $C^I$, the generalized local homology $\text{H}^I_i(M,N)$ belongs to $\mathcal{S}$ for all $i>pd(M)$. Also, if $\mathcal{S}$ satisfies the condition $C_I$, then the generalized local cohomology module $\text{H}^i_I(M,N)\in \mathcal{S}$ for all $i>pd(M)$.