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 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).
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 (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 .)
For example, if [2] 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,
- the r-th Chow group of X.