nLab rational equivalence

Let $X$ be a noetherian scheme. One defines a notion of rational equivalence on $X$ which, roughly speaking, identifies algebraic cycles that are connected by a family of cycles parametrized by the projective line.

A cycle $C \in Z_k(X)$ is rationally equivalent to zero if there are closed integral subschemes $Z_\alpha$ of dimension $k + 1$ and invertible rational functions $r_\alpha \in R(Z_\alpha)^*$ such that

$$C = \sum_\alpha j_{\alpha,*}(div(r_\alpha)) \,,$$

where

• $div$ denote the associate Weil divisors;

• $j_{\alpha,*}$ denotes the direct image along the inclusion $Z_\alpha \hookrightarrow X$.

Rational equivalence generalizes linear equivalence? of Weil divisors. It is an example of an adequate equivalence relation.

References

• Stacks Project, 02RV