# 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

Last revised on May 30, 2013 at 16:09:50. See the history of this page for a list of all contributions to it.