---
tag: 95N3QVHAQN
kind: definition
name: Rational equivalence
msc: [14C15, 14C25]
defines: K76N026JFK
uses: [9VFWPZEVGN, HV6YBS0YDH, 2D5SS6WM14, V6JBKT168T]
sources:
  - url: https://en.wikipedia.org/wiki/Chow_group
---

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field, and let $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$.
Let $W \subset X$ be an irreducible closed subvariety of dimension $r + 1$, and let $f$ be a nonzero element of its [[function field|2D5SS6WM14]] $k(W)$.
For each irreducible closed subvariety $V \subset W$ of dimension $r$, the local ring $A = \mathcal{O}_{W, V}$ is a one-dimensional Noetherian local domain with fraction field $k(W)$.
Write $f = a/b$ with $a, b \in A \setminus \set{0}$.
Set $\operatorname{ord}_V(f) = \ell_A(A/aA) - \ell_A(A/bA)$, where $\ell_A$ is the [[length|V6JBKT168T]] of an $A$-module.
This does not depend on the choice of $a$ and $b$.
The *divisor* of $f$ is the $r$-cycle $\operatorname{div}(f) = \sum_V \operatorname{ord}_V(f) \, [V]$, a finite sum.
Let $\operatorname{Rat}_r(X) \subset Z_r(X)$ be the subgroup generated by all such $\operatorname{div}(f)$, for all $W$ and $f$.
Two $r$-cycles $Z, Z'$ are *rationally equivalent* if $Z - Z' \in \operatorname{Rat}_r(X)$.

## Remarks

When $W$ is normal along $V$, $\operatorname{ord}_V$ is the discrete valuation of $k(W)$ at $V$.
Example: on $X = \mathbb{P}^1$ over a field $k$, take $W = X$ and $f = (x - a)/(x - b)$ with $a \ne b$ in $k$.
Then $\operatorname{div}(f) = [a] - [b]$, so the points $a$ and $b$ are rationally equivalent.
Non-example: on an elliptic curve $E$ over an algebraically closed field, two distinct points $P$ and $Q$ are not rationally equivalent.
If they were, then $[P] - [Q] = \operatorname{div}(f)$ for some $f \in k(E)^{\times}$, since $W = E$ is the only choice on a curve.
Then $f$ would be a rational function of degree $1$, hence an isomorphism $E \to \mathbb{P}^1$, which is impossible because $E$ has genus $1$.
Equivalently, $Z - Z' = \sum_i \big([V_i(0)] - [V_i(\infty)]\big)$ for finitely many subvarieties $V_i \subset X \times \mathbb{P}^1$ of dimension $r + 1$ that map onto $\mathbb{P}^1$, where $V_i(a)$ is the fiber of $V_i$ over $a$, viewed in $X$.