Product structures in motivic cohomology and higher Chow groups

It is shown that the product structures of motivic cohomology groups and of higher Chow groups are compatible under the comparison isomorphism of Voevodsky (2002) [11], . This extends the result of Weibel (1999) [14], where he used the comparison isomorphism which assumed that the base field admits resolution of singularities.The mod n motivic cohomology groups and product structures in motivic homotopy theory are defined, and it is shown that the product structures are compatible under the comparison isomorphisms.