Algebraic cycle

A $k$-cycle on a scheme $X$ is a finite integer linear combination of integral closed subschemes of dimension $k$. Their free abelian group is denoted $Z_k(X)$.

Rational equivalence is generated by principal divisors $[\mathrm{div}(r)]$ of nonzero rational functions on integral subvarieties one dimension larger than the cycles.

The Chow group is $A_k(X)=Z_k(X)/{\mathrm{Rat}}_k(X)$, the group of $k$-dimensional algebraic cycles modulo rational equivalence.

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

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.

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 variety, intersection of cycles gives the graded Chow ring $A^{\ast }(X)$.

For smooth projective varieties $X,Y$, pull back Chow classes from the two factors and take their intersection product on $X\times Y$. This gives the external product and a graded map ${\mathrm{CH}}^{\ast }(X)\otimes {\mathrm{CH}}^{\ast }(Y)\to {\mathrm{CH}}^{\ast }(X\times Y)$. Unlike the cohomological Künneth theorem, it need not generate every class of the product.

On $E\times E$ for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.

For a closed immersion $j:Z\hookrightarrow X$ with open complement $i:U\hookrightarrow X$, restriction and proper pushforward give an exact sequence

A cellular decomposition filters a scheme by closed subschemes whose successive differences are disjoint unions of affine spaces. The closures of the cells generate its Chow groups.

For the hyperplane class $H$,and $A_k({\mathbb{P}}^r)\cong \mathbb{Z}$ is generated by a linear ${\mathbb{P}}^k$.

For a rank-$e+1$ vector bundle $E\to X$ and $p:\mathbb{P}(E)\to X$, powers of $\zeta =c_1(\mathcal{O}(1))$ give isomorphisms

For a rank-$r$ vector bundle $E\to X$, flat pullback is an isomorphism $A_k(X)\cong A_{k+r}(E)$.