nLab
Chow group

Contents

Idea

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

Definition

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

CH k(X)Z k(X)/ rat 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. num, hom, alg\sim_{num}, \sim_{hom}, \sim_{alg}) in place of rational equivalence, to get groups

CH k (X)=Z k(X)/ 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.

References

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:

Informal lecture notes by Jacob Murre?:

  • Jacob Murre?, Lectures on algebraic cycles and Chow groups. Summer school on Hodge theory and related topics, ICTP, 2010. PDF

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

Revised on May 30, 2013 13:14:03 by Urs Schreiber (82.113.121.121)