Motivic cohomology are isomorphic to higher Chow groups, by Vladimir Voevodsky

In this short paper we show that the motivic cohomology groups defined in "Cycles, transfers and motivic homology theories" (No. 368 of this archive) are isomorphic to the higher Chow groups for smooth varieties over any field. Because of the homotopy invariance property of higher Chow groups this implies that the Cancellation Theorem holds over any field. Both facts were previously known only under the resolution of singularities assumption.

Vladimir Voevodsky <>