nLab
adequate equivalence relation

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 $𝒵={𝒵}_{*}$ 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 ${𝒵}_{*}$.

Let ${\mathrm{SmProj}}_k$ be the category of smooth projective varieties over $k$. A rule giving an equivalence relation ${𝒵}^{*}(X)$ 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 $a,b\in {𝒵}^{*}(X)$, there exists $a\prime \sim a$ such that $a\prime$ 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 $a\in {𝒵}^{*}(X)$ and $b\in {𝒵}^{*}(X\times Y)$ 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 ${𝒵}^{*}$, then becomes globally defined on ${𝒵}^{*}/\sim$.

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.