Framework classified in a linear algebraic group

We give an abstract version of Totaro's definition of the Chow groups of the classifying space of a linear algebraic group G over a field k, so that it yields automatically a definition of F(BG) for functors F satisfying some simple axioms. If F is kale motivic cohomology, k is algebraically closed and G is finite, F(BG) essentially boils down to the integral cohomology of G. In general, we get coniveau spectral sequences converging to the kale motivic cohomology of BG, which unify and generalize invariants of G previously considered by Bogomolov and Serre. (C) 2014 Elsevier Masson SAS. Tous droits reserves.