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The equivalence class of an algebraic cycle modulo rational equivalence. The Chow group consists of these classes with addition induced by adding cycles.

The cycle class of an algebraic cycle is its image under the cycle class map. For a codimension-$r$ cycle on a smooth complex projective variety, it belongs to $H^{2r}(X,\mathbb{Z})$ and is the Poincare dual of its fundamental homology class. The map factors through the Chow group, but a Chow class retains finer information than its cohomological cycle class.

For a smooth complex projective algebraic variety, the cycle class map sends an algebraic cycle of codimension $r$ to the Poincare dual of its homology class. Rational equivalence gives the same cohomology class, so the map descends to the Chow group. It respects intersection products, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of Chow classes.

Restriction of cycles from $X$ to $U$ is surjective: every integral subvariety of $U$ is the restriction of its closure in $X$. Its kernel on cycle groups consists exactly of cycles supported on $Z$, hence is the image of $Z_k(Z)\to Z_k(X)$.

This descends to the Chow group level. If a cycle $\alpha$ restricts to zero in $A_k(U)$, express its restriction as a sum of principal divisors on $(k+1)$-dimensional subvarieties of $U$. Taking their closures in $X$ and the same rational functions gives a rationally equivalent cycle whose difference from $\alpha$ is supported on $Z$. Thus $[\alpha ]$ lies in the image of $A_k(Z)$. Since principal divisors restrict to principal divisors, the other composite is zero. This proves the localization sequence for Chow groups

At the generic point ${\eta }_V$ of $V$, the local ring $R={\mathcal{O}}_{W,{\eta }_V}$ is a one-dimensional Noetherian local domain. Write $r=a/b$ with nonzero $a,b\in R$ and define the order of vanishingThis is independent of the representation. The associated principal $k$-cycle iswhere only finitely many terms are nonzero.

The subgroup ${\mathrm{Rat}}_k(X)\subseteq Z_k(X)$ is generated by these cycles as $W$ ranges over integral $(k+1)$-dimensional subvarieties and $r$ over $R(W{)}^{\times }$. Two $k$-cycles are rationally equivalent when their difference lies in this subgroup, and the Chow group is

For a proper morphism of varieties $f:X\to Y$, a Cartier divisor $D$ on $Y$, and an algebraic cycle $\alpha$ on $X$, pullback of $D$, intersection with $\alpha$, and proper pushforward satisfy the displayed identity in the Chow group. In particular, for a generically finite morphism of surfaces of degree $r$, $(f^{\ast }D{)}^2=r{D}^2$. For a curve $C$ contracted to a point, $f_{\ast }C=0$, so $f^{\ast }D\cdot C=0$.