Title: Homology of linear groups via cycles in BG x X
Author1: Kevin P. Knudson 
Author2: Mark E. Walker  
email1: knudson@math.msstate.edu 
email2: mwalker@math.unl.edu 

Abstract:
Let G be an algebraic group and let X be a smooth integral scheme over a
field k.  In this paper we construct homology-type groups H_i(X,G) by
considering cycles in the simplicial scheme BG x X (an idea suggested by
Andrei Suslin).  We discuss the basic properties of these groups and
construct a spectral sequence, beginning with the groups
H_i(\Delta^j,G), which converges to the etale cohomology of the
simplicial group BG.  These groups are therefore connected with the
study of Friedlander's generalized isomorphism conjecture.  We also
compute some examples, focusing in particular on the case X=Spec(k).  In
the case where k is the real numbers, there is a connection between the
groups H_i and the Z/2-equivariant cohomology of the classifying space
of the discrete group G(R).