Mixed Motives

base scheme S, whose construction is based on the rough ideas I originally outlined

in a lecture at the J.A.M.I. conference on K-theory and number theory, held at the

Johns Hopkins University in April of 1990.The essential principle is that one can

form a categorical framework for motivic cohomology by first forming a tensor category from the category of smooth quasi-projective schemes over S, with morphisms

generated by algebraic cycles, pull-back maps and external products, imposing the

relations of functoriality of cycle pull-back and compatibility of cycle products with

the external product, then taking the homotopy category of complexes in this tensor

category, and finally localizing to impose the axioms of a Bloch-Ogus cohomology

theory, e.g., the homotopy axiom, the K¨unneth isomorphism, Mayer-Vietoris, and

so on.

Remarkably, this quite formal construction turns out to give the same cohomology theory as that given by Bloch’s higher Chow groups [19], (at least if the

base scheme is Spec of a field, or a smooth curve over a field).In particular, this

puts the theory of the classical Chow ring of cycles modulo rational equivalence in

a categorical context.

Following the identification of the categorical motivic cohomology as the higher

Chow groups, we go on to show how the familiar constructions of cohomology:

Chern classes, projective push-forward, the Riemann-Roch theorem, Poincar´e duality, as well as homology, Borel-Moore homology and compactly supported cohomology, have their counterparts in the motivic category.The category of Chow

motives of smooth projective varieties, with morphisms being the rational equivalence classes of correspondences, embeds as a full subcategory of our construction.

Our motivic category is specially constructed to give realization functors for

Bloch-Ogus cohomology theories.As particular examples, we construct realization

functors for classical