We extend the Torsion Sub-complex Reduction technique to the case where the kernel of a group action is non-trivial. For this purpose, we excise the non-central torsion sub-complex from the equivariant spectral sequence. This new method of excision in equivariant cohomology is elaborated in a general setting. We demonstrate the power of the method by establishing general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL_2 groups over imaginary quadratic integers on their associated symmetric space.