A characterization result of cofinite local cohomology modules

Let $R$ be a commutative Noetherian ring, $\text{fa}$ an ideal of $R$, $M$ an arbitrary $R$-module and $N$ a finite $R$-module. We prove that \cite[Theorem 2.1]{Mel} and \cite[Proposition 3.3 (i)$\iff$(ii)]{B1} are true for any Serre subcategory of $R$-modules. We also prove a characterization theorem for ${\text{lc}}_{\text{fa}}^i(M)$ and ${\text{lc}}_{\text{fa}}^i(N,M)$ to be $\text{fa}$-cofinite for all $i$, whenever one of the following cases holds: (a) $\text{ara}(\text{fa})\le 1$, (b) $\dim R\mathrm{/}\text{fa}\le 1$ or © $\dim R\le 2$. In the end we study Artinianness and Artinian $\text{fa}$-cofiniteness of local cohomology modules.