In algebraic geometry, Bloch's higher Chow groups, a generalization of Chow group, is a precursor and a basic example of motivic cohomology (for smooth varieties).
It was introduced by Spencer Bloch (Bloch 1986) and the basic theory has been developed by Bloch and Marc Levine.
In more precise terms, a theorem of Voevodsky[1] 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.
One of the motivations for higher Chow groups comes from homotopy theory.
α , β ∈
are algebraic cycles in
which are rationally equivalent via a cycle
, and the higher Chow groups are meant to encode the information of higher homotopy coherence.
can be thought of as the homotopy classes of cycles while
can be thought of as the homotopy classes of homotopies of cycles.
Let X be a quasi-projective algebraic scheme over a field (“algebraic” means separated and of finite type).
, define which is an algebraic analog of a standard q-simplex.
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
is an effective Cartier divisor, there is the Gysin homomorphism: that (by definition) 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: (More simply, since
is naturally a simplicial abelian group, in view of the Dold–Kan correspondence, higher Chow groups can also be defined as homotopy groups
[2] is a closed subvariety such that the intersections
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, Proper maps
are covariant between the higher chow groups while flat maps are contravariant.
is smooth, any map to
is an algebraic vector bundle, then there is the homotopy equivalence
Given a closed equidimensional subscheme
there is a localization long exact sequence
In particular, this shows the higher chow groups naturally extend the exact sequence of chow groups.
(Bloch 1994) showed that, given an open subset
has pure codimension, then it yields the long exact sequence for higher Chow groups (called the localization sequence).