The Motivic Dga

The main goal of this paper is to associate to each smooth quasi-projective scheme X over any field an E∞ differential graded algebra whose cohomology groups are the higher Chow groups of X. Modulo torsion, this provides a strictly commutative differential graded algebra. Such a construction is currently known only for the case X itself is a field. Needless to say there are several applications of this result, some of which are considered here: for example, we are able to construct a category of relative mixed Tate-motives associated to any such scheme under the hypothesis that the DGA we obtain is connected. We also show that this holds for all smooth linear projective varieties over a field k, if the Beilinson-Soulé vanishing condition holds for the field. We also establish certain cohomology operations on mod-p motivic cohomology by this procedure.