nLab
Chow group

Idea

The Chow group of a scheme $X$ is the analog of the singular chain homology? group of a topological space.

Definition

There is a notion of algebraic cycle? $\in Z_i(X)$ generalizing that of a singular cycle. One first introduces some adequate equivalence relation $\sim$ on the set of cycles. The main example is the rational equivalence of cycles.

The $i$th Chow group ${\mathrm{CH}}_i(X):=Z_i(X{)}_{\sim }$ of $X$ is the group of equivalence classes of algebraic cycles in $X$.

The total Chow group is the direct sum of all these

$$\mathrm{CH}(X):={\oplus }_i{Z}_i(X{)}_{\sim }.$$CH(X) := \oplus_i Z_i(X)_\sim \,.

Cohomological interpretation

Chow groups appear as the cohomology groups of motivic cohomology (see there for details) with coefficients in suitable Eilenberg-MacLane objects.

References

A concise definition of the notion of Chow group and related concepts is in

• this pdf from some Vigre minicours (???)