nLab Chow group

Contents

Idea
Definition
Cohomological interpretation
Related concepts
References

Idea

The Chow groups of a noetherian scheme $X$ are the analogs of the singular homology groups of a topological space.

Definition

Let $X$ be a noetherian scheme. One defines the $k$ -th Chow group of $X$ as the quotient of the group $Z_k(X)$ of algebraic cycles of dimension $k$ by the subgroup of algebraic cycles rationally equivalent to zero:

CH_k(X) \coloneqq Z_k(X) / \sim_{\rat}

The Chow ring is the graded ring which is the direct sum of the Chow groups, with multiplication being the intersection product.

More generally one can use any adequate equivalence relation $\sim$ (e.g. $\sim_{num}, \sim_{hom}, \sim_{alg}$ ) in place of rational equivalence, to get groups

CH^{\sim}_k(X) = Z_k(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.

Related concepts

References

Named after Wei-Liang Chow.

The canonical reference is

William Fulton, Intersection theory, 1998. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2, Berlin, New York: Springer-Verlag

The original references are

Pierre Samuel?, Rational Equivalence of Arbitrary Cycles. American Journal of Mathematics, Vol. 78, No. 2 (Apr., 1956), pp. 383-400
Claude Chevalley, Les classes d’equivalence rationnelles I. Séminaire Claude Chevalley, 3 (1958), Exp. No. 2, 14 (on NUMDAM)
Claude Chevalley, Les classes d’équivalence rationnelle, II. Séminaire Claude Chevalley, 3 (1958), Exp. No. 3, 18 (on NUMDAM)

The most general treatment can be found in the The Stacks Project:

Aise Johan de Jong et. al., The Stacks Project, chapter 38, Chow Homology and Chern Classes (URL)