---
tag: K76N026JFK
kind: concept
name: Rational equivalence
msc: [14C15, 14C25]
sources:
  - url: https://en.wikipedia.org/wiki/Chow_group
---

## Summary

Rational equivalence is the finest of the usual equivalence relations on [[algebraic cycles|9VFWPZEVGN]].
Two cycles are rationally equivalent when their difference is a sum of divisors of rational functions on subvarieties.
The quotient of the group of cycles by rational equivalence is the [[Chow group|EBEF6PEK4N]].

## Explanations

### L1 human

On a curve, the zeros of a rational function minus its poles form a cycle of points that one wants to count as zero.
Rational equivalence declares all such cycles to be zero, and does the same on every subvariety of every dimension.
On the projective line, the function $(x - a)/(x - b)$ has a zero at $a$ and a pole at $b$, so any two points are rationally equivalent.
On an elliptic curve, two distinct points are never rationally equivalent, so the relation sees the geometry of the curve.