---
tag: EBEF6PEK4N
kind: concept
name: Chow group
msc: [14C15, 14C25]
sources:
  - url: https://en.wikipedia.org/wiki/Chow_group
same_as:
  wikidata: Q5105523
  wikipedia: Chow group
---

## Summary

The Chow groups of a variety $X$ are the groups of [[algebraic cycles|9VFWPZEVGN]] on $X$ modulo [[rational equivalence|K76N026JFK]], one for each dimension.
They are the algebraic analogue of homology groups.
For a smooth variety, the intersection product turns their sum into a graded ring, the Chow ring, which is the setting of [[intersection theory|4HV4WTAX9P]].
In codimension one they recover the divisor class group and the [[Picard group|3Z9DCWYXQZ]] of a smooth variety.

## Explanations

### L1 human

A Chow group counts subvarieties of a given dimension, where two of them count as the same when one can be deformed into the other through a rational family.
On the projective plane, every curve of degree $d$ is equivalent to $d$ times a line, so the group of curves is $\Z$, generated by the class of a line.
On an elliptic curve, the group of points is much bigger: it records the degree and also a point of the curve itself.
The Chow groups are thus simple for rational varieties and can be enormous in general.