An algebraic cycle on a scheme $X$ of finite type over a field $k$ is a finite linear combination $\sum_{i=1}^r n_i Z_i$ of integral closed subschemes $Z_i\subset X$ with integral coefficients $n_i$. The algebraic cycles form a group $\mathcal{Z}= \mathcal{Z}_*$ of algebraic cycles on $X$ which is graded by the dimensions of the cycles. Sometimes (for equidimensional $X$) one looks at the grading by codimension $\mathcal{Z}_*$.

Let $\mathrm{SmProj}_k$ be the category of smooth projective varieties over $k$. A rule giving an equivalence relation $\mathcal{Z}^*(X)$ for every $X$ in $\mathrm{SmProj}_k$, and which is compatible with grading is an adequate equivalence relation such that

1. For every pair of cycles $a,b\in \mathcal{Z}^*(X)$, there exists $a'\sim a$ such that $a'$ is transversal to $b$.

2. Consider a product $X\times Y$ in $\mathrm{SmProj}_k$, denote by $p_X,p_Y$ its projections. Consider cycles $a\in \mathcal{Z}^*(X)$ and $b\in \mathcal{Z}^*(X\times Y)$ such that $b$ and $p_X^*(a)$ intersect properly. Then $a\sim 0$ implies $p_Y_*(p_X^*(a)\cdot b) \sim 0$ where $p_X^*(a)\cdot b$ denotes the intersection product.

The intersection product, which is associative but only partially defined on $\mathcal{Z}^*$, then becomes globally defined on $\mathcal{Z}^*/{\sim}$.

Typical choices are rational, algebraic and numerical adequate equivalence relations. The rational is the finest and the numerical is the coarsest nonzero adequate equivalence relation.

Revised on May 28, 2009 21:46:43 by Toby Bartels (71.104.230.172)