Chow ring

The ring of rational equivalence classes of algebraic cycles (cf. Algebraic cycle) on a non-singular quasi-projective algebraic variety. Multiplication in this ring is defined in terms of intersections of cycles (cf. Intersection theory).

The Chow ring of a variety is a graded commutative ring, where denotes the group of classes of cycles of codimension . For a morphism the inverse-image homomorphism is a homomorphism of rings, and the direct-image homomorphism is (for proper ) a homomorphism of -modules. This means that there is a projection formula:

The Chow ring is the domain of values for the theory of Chern classes of vector bundles (cf. [1]). More precisely, if is a locally trivial sheaf of rank over a variety , if is its projectivization, if is the canonical projection, and if is the class of divisors corresponding to the invertible sheaf , then is an imbedding and the Chow ring may be identified with the quotient ring of the polynomial ring by the ideal generated by the polynomial

The coefficient is called the -th Chern class of the sheaf .

In the case of a variety over the field of complex numbers, there is a homomorphism into the singular cohomology ring that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.

If is a singular quasi-projective variety, then its Chow ring is defined as the direct limit of rings over all morphisms , where is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. [3]).

References

[1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[2]	"Anneaux de Chow et applications" , Sem. Chevalley (1958) Zbl 0098.13101
[3]	W. Fulton, "Rational equivalence on singular varieties" Publ. Math. IHES , 45 (1975) pp. 147–167 MR0404257 Zbl 0332.14002

Comments

For a Noetherian scheme (or ring), let denote the -groups of (the category of) finitely-generated projective modules over ; cf. Algebraic -theory. Let denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf where runs through the open (affine) subschemes of . One then has the Bloch formula [a1]

providing a link between the Chow groups of and the cohomology of with values in the -sheaves of . Using results on the algebraic -theory of fields, [a2], this can be used to obtain results on the , in particular , [a3]. Another often used notation for the Chow group is instead of .

Cf. Sheaf theory for the notions of sheafification, pre-sheaf, sheaf, and cohomology with values in a sheaf.

References

[a1]	S. Bloch, "Lectures on algebraic cycles" , Dept. Math. Duke Univ. (1980) MR0558224 Zbl 0436.14003
[a2]	A.S. Merkur'ev, A.A. Suslin, "-cohomology of Severi–Brauer varieties and norm residue homomorphism" Math. USSR Izv. , 21 (1983) pp. 307–340 Izv. Akad. Nauk SSSR Ser. Mat. , 46 : 5 (1982) pp. 1011–1046 Zbl 0525.18008 Zbl 0525.18007
[a3]	J.-L. Colliot-Thélène, "Hilbert's theorem 90 for with application to the Chow groups of rational surfaces" Inv. Math. , 71 (1983) pp. 1–20
[a4]	W. Fulton, "Intersection theory" , Springer (1984) MR0735435 MR0732620 Zbl 0541.14005