upper bounds for finiteness of generalized local cohomology modules

let $r$ be a commutative noetherian ring with non-zero identity and $fa$ an ideal of $r$. let $m$ be a finite $r$--module of finite projective dimension and $n$ an arbitrary finite $r$--module. we characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(m,n)$ 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 $n > pd m$ be an integer such that $lc^{i}_{fa}(m,n)$ belongs to $mathcal s$ for all $i> n$. then, for any ideal $fbsupseteq fa$, it is also shown that the module $lc^{n}_{fa}(m,n)/{fb}lc^{n}_{fa}(m,n)$ belongs to $mathcal s$.