depth of an ideal on a pair of modules

‎let r be a commutative noetherian ring and i an ideal of r‎. ‎suppose that s is a serre subcategory of the category of r-modules which satisfies the condition ci‎. ‎let m be a zd-module and n an r-module‎. ‎as a generalization of the notion of s-depth(i‎, ‎m)‎, ‎we define the s-depth of i ‎on the pair (n‎, ‎m) by s-depth(i‎, ‎n‎, ‎m):=s-depth(annr(n/in)‎, ‎m)‎. ‎we investigate the connections between s-depth(i‎, ‎n‎, ‎m)‎, ‎local cohomology modules and ext functors‎. ‎in particular‎, ‎when n is finitely generated‎, ‎it is shown that‎ ‎s-depth(i‎, ‎n‎, ‎m)=inf {i‎: ‎h{i}i(n‎, ‎m) ∉ s}=inf{i‎: extir(n/{in}‎, ‎m)∉ s}‎. ‎moreover‎, ‎various formulas are provided that relate this generalized s-depth to other notions of depth in the literature‎.