# Chow ring

2010 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]).

How to Cite This Entry:
Chow ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chow_ring&oldid=25308
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article