Upper bounds for finiteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity and $\text{fa}$ an ideal of $R$. Let $M$ be a finite $R$–module of of finite projective dimension and $N$ an arbitrary finite $R$–module. We characterize the membership of the generalized local cohomology modules ${\text{lc}}_{\text{fa}}^i(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal{S}$ be a Serre subcategory of the category of $R$–modules and $n⩾\text{pd}M$ be an integer such that ${\text{lc}}_{\text{fa}}^i(M,N)$ belongs to $\mathcal{S}$ for all $i>n$. If $\text{fb}$ is an ideal of $R$ such that ${\text{lc}}_{\text{fa}}^n(M,N\mathrm{/}\text{fb}N)$ belongs to $\mathcal{S}$, It is also shown that the module ${\text{lc}}_{\text{fa}}^n(M,N)\mathrm{/}\text{fb}{\text{lc}}_{\text{fa}}^n(M,N)$ belongs to $\mathcal{S}$.