Bloch's higher Chow group
In algebraic geometry, Bloch's higher Chow groups, a generalization of Chow group, is a precursor and a basic example of motivic cohomology. It was introduced by Spencer Bloch and the basic theory has been developed by Bloch and Marc Levine.
In more precise terms, a theorem of Voevodsky implies: for a smooth scheme X over a field and integers p, q, there is a natural isomorphism
between motivic cohomology groups and higher Chow groups.
Motivation
One of the motivations for higher Chow groups comes from homotopy theory. In particular, if are algebraic cycles in which are rationally equivalent via a cycle, then can be thought of as a path between and, and the higher Chow groups are meant to encode the information of higher homotopy coherence. For example,can be thought of as the homotopy classes of cycles whilecan be thought of as the homotopy classes of homotopies of cycles.Definition
Let X be a quasi-projective algebraic scheme over a field.For each integer, define
which is an algebraic analog of a standard q-simplex. For each sequence, the closed subscheme, which is isomorphic to, is called a face of.
For each i, there is the embedding
We write for the group of algebraic i-cycles on X and for the subgroup generated by closed subvarieties that intersect properly with for each face F of.
Since is an effective Cartier divisor, there is the Gysin homomorphism:
that maps a subvariety V to the intersection
Define the boundary operator which yields the chain complex
Finally, the q-th higher Chow group of X is defined as the q-th homology of the above complex:
For example, if is a closed subvariety such that the intersections with the faces are proper, then
and this means, by Proposition 1.6. in Fulton’s intersection theory, that the image of is precisely the group of cycles rationally equivalent to zero; that is,
Properties
Functoriality
Proper maps are covariant between the higher chow groups while flat maps are contravariant. Also, whenever is smooth, any map to is contravariant.Homotopy invariance
If is an algebraic vector bundle, then there is the homotopy equivalenceLocalization
Given a closed equidimensional subscheme there is a localization long exact sequencewhere. In particular, this shows the higher chow groups naturally extend the exact sequence of chow groups.Localization theorem
showed that, given an open subset, for,is a homotopy equivalence. In particular, if has pure codimension, then it yields the long exact sequence for higher Chow groups.