Homology of linear groups via cycles in BG× X

Abstract

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 H i(X,G) from cycles in the simplicial scheme BG× X. When X=Spec( k) and k is algebraically closed, these groups are the ordinary homology groups H i ( 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 X= Spec( R) .