Homology of linear groups via cycles in BG×X

For X a smooth separated scheme of finite type over a field k, and for G a linear algebraic group over k, we construct homology-type functors image from cycles in the simplicial scheme BG×X. When X=Spec(k) and k is algebraically closed, these groups are the ordinary homology groups Hi(G(k)) of the discrete group of k-rational points of G. We construct a spectral sequence beginning with our groups and converging to the étale cohomology of the simplicial scheme BG, thus relating this theory to conclude with some calculations in the case where image.