Relative left derived functors of tensor product functors

We introduce and study the relative left derived functor Tor n (F,F.)(.,.) in the module category, which unifies several related left derived functors. Then we give some criteria for computing the F-resolution dimensions of modules in terms of the properties of TorTor n (F,F.)(.,.). We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.