generalized local cohomology and serre cohomological dimension

‎let $r$ be a commutative noetherian ring, $i,~j$ be two ideals of $r$, and  $m,~n$ be two $r$-modules. let $s$ be a serre subcategory of the category of $r$-modules.we introduce serre cohomological dimension of $n, m$ with respect to $(i,j)$, as${rm cd}_s(i, j, n, m)=sup{iin bbb n_0: h_{i,j}^{i}(n, m)notin s}.$we study some properties of ${rm cd}_s(i, j, n, m)$, and we getsome formulas and upper bounds for it.