Chow ring

$ \newcommand{\CH}{\mathrm{CH}} $

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 $\CH(X)=\bigoplus_{i\geq 0} \CH^i(X)$ of a variety $X$ is a graded commutative ring, where $\CH^i(X)$ denotes the group of classes of cycles of codimension $i$. In earlier publications, the notation $\mathrm{A}(X)$ is sometimes used instead of $\CH(X)$.

For a morphism $f:X \to Y$ the inverse-image homomorphism $f^*:\CH(Y) \to \CH(X)$ is a homomorphism of rings, and for $f$ proper, the direct-image homomorphism $f_*: \CH(X)\to \CH(Y)$ is a homomorphism of $\CH(Y)$-modules. This means that there is a projection formula:

$$f_*(f^*(y)\cdot x) = y\cdot f_*(x), \quad x \in CH(X), \quad y \in CH(Y)$$

The Chow ring is the domain of values for the classical theory of Chern classes of vector bundles (cf. [1]). More precisely, if $E$ is a locally free sheaf of rank $r$ over a variety $X$, if $\pi:P(E) \to X$ is its projectivization and if $\zeta \in \CH^1(P(E))$ is the class of the divisor corresponding to the invertible sheaf $\mathcal{O}_{P(E)}(1)$, then $\pi^*$ is injective and the Chow ring $\CH(P(E))$ may be identified with the quotient ring of the polynomial ring $\CH(X)[\zeta]$ by the ideal generated by the polynomial

$$\zeta^r -c_1(E)\zeta^{r-1}+\cdots + (-1)^r c_r(E).$$

The coefficient $c_k(E)\in \CH^k(X)$ is called the $k$-th Chern class of the locally free sheaf $E$.

In the case of a variety over the field of complex numbers, there is a homomorphism $\CH(X) \to \mathrm{H}(X,\mathbb Z)$ into the singular cohomology ring that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.

If $X$ is a singular quasi-projective variety, then its Chow ring $\CH(X)$ is defined as the direct limit of rings $\CH(X)=\varinjlim \CH(Y)$ over all morphisms $f:X \to Y$, where $Y$ 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 $X$ a Noetherian scheme (or ring), let $K_n(X)$ denote the $n$-th $K$-group of (the category of) finitely-generated projective modules over $X$; cf. Algebraic K-theory. Let $\mathcal{K}$ denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf $U \mapsto K_n(U)$ where $U$ runs through the open (affine) subschemes of $X$. One then has the Bloch formula [a1]

$$\CH^p(X) \simeq H^p(X, \mathcal{K}_p)

providing a link between the Chow groups of $X$ and the cohomology of $X$ with values in the $\mathcal{K}$-sheaves of $X$. Using results on the algebraic K-theory of fields, [a2], this can be used to obtain results on Chow groups, in particular on $\CH^2$, [a3].

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, "K-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 $K_2$ 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