Mixed Motives
Marc Levinebase 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
…