Some results on generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\text{fa}$ an ideal of $R$, $M$ a finite $R$–module and $X$ an arbitrary $R$–module. Here, we show that, in the Serre subcategories of the category of $R$–modules, how the generalized local cohomology modules, the ordinary local cohomology modules and the extension modules behave similarly at the initial points. We conclude some Artinianness and cofiniteness results for ${\text{lc}}_{\text{fa}}^n(M,X)$, and some finiteness results for ${\text{Supp}}_R({\text{lc}}_{\text{fa}}^n(M,X))$ and ${\text{Ass}}_R({\text{lc}}_{\text{fa}}^n(M,X))$.