Motivic cohomology

Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.

Let X be a scheme of finite type over a field k. A key goal of algebraic geometry is to compute the Chow groups of X, because they give strong information about all subvarieties of X. The Chow groups of X have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme Z of X, there is an exact sequence of Chow groups, the localization sequence

whereas in topology this would be part of a long exact sequence.

This problem was resolved by generalizing Chow groups to a bigraded family of groups, (Borel–Moore) motivic homology groups (which were first called higher Chow groups by Bloch). Namely, for every scheme X of finite type over a field k and integers i and j, we have an abelian group H_i(X,Z(j)), with the usual Chow group being the special case

For a closed subscheme Z of a scheme X, there is a long exact localization sequence for motivic homology groups, ending with the localization sequence for Chow groups:

In fact, this is one of a family of four theories constructed by Voevodsky: motivic cohomology, motivic cohomology with compact support, Borel-Moore motivic homology (as above), and motivic homology with compact support. These theories have many of the formal properties of the corresponding theories in topology. For example, the motivic cohomology groups Hⁱ(X,Z(j)) form a bigraded ring for every scheme X of finite type over a field. When X is smooth of dimension n over k, there is a Poincare duality isomorphism

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