Generic motives and motivic cohomology of fields

These notes are devoted to the study of generic motives and their application to the structure of the motivic cohomology of fields. Related to Voevodsky's homotopy t-structure on motivic complexes, generic motives are also central to the conjectural motivic t-structure, as was shown by Beilinson. Morphisms of generic motives are closely related to the functoriality of Rost's cycle modules. e obtain some new computations of generic motives, and deduce computations of morphisms of generic motives, which draw a conjectural picture. The structure of generic motives reflects into that of motivic cohomology of fields. After giving a new argument for determining weights in the K-theory of number fields using Borel's isomorphism, we deduce that many motivic cohomology groups of fields have infinite rank, usually of the same cardinality as the field itself. As an example motivic cohomology groups of the real or complex numbers are shown to be uncountable in many degrees and twists. Despite these results, we formulate a hope on their behavior with respect to transcendental elements, providing a counterpart to the Beilinson-Soul\'e vanishing conjecture.