Chow ring

2020 Mathematics Subject Classification: Primary: 14Cxx Secondary: 14G1018F25 [MSN][ZBL]

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

The Chow ring of a non-singular quasi-projective algebraic variety is the ring of rational equivalence classes of algebraic cycles (cf. Algebraic cycle) on that 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. [Ha]). 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. [Fu]).

References

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

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 [Bl]

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

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, [MeSu], this can be used to obtain results on Chow groups, in particular on $\CH^2$, [Co].

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

References

[Bl]	S. Bloch, "Lectures on algebraic cycles", Dept. Math. Duke Univ. (1980) MR0558224 Zbl 0436.14003
[Co]	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 MR0688259 Zbl 0527.14011
[Fu2]	W. Fulton, "Intersection theory", Springer (1984) MR0735435 MR0732620 Zbl 0541.14005
[MeSu]	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