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.

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

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 $A_k(X)\cong A_{k+r}(E)$.