---
tag: W6V340HYNK
kind: definition
name: Chow group
msc: [14C15, 14C25]
defines: EBEF6PEK4N
uses: [9VFWPZEVGN, K76N026JFK, HV6YBS0YDH]
sources:
  - url: https://en.wikipedia.org/wiki/Chow_group
---

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field and let $r \ge 0$ be an integer.
Let $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$ and $\operatorname{Rat}_r(X)$ its subgroup of cycles [[rationally equivalent|K76N026JFK]] to $0$.
The *Chow group* of $r$-cycles on $X$ is the quotient
$$CH_r(X) = Z_r(X) / \operatorname{Rat}_r(X).$$
If $X$ is equidimensional of dimension $n$, the Chow group of codimension $j$ cycles is $CH^j(X) = CH_{n-j}(X)$.

## Remarks

Example: for $X = \mathbb{P}^n$ over a field and $0 \le r \le n$, $CH_r(\mathbb{P}^n) \cong \Z$, generated by the class of a linear subspace of dimension $r$.
A subvariety of dimension $r$ and degree $d$ has class $d$ times the generator.
For $r = 0$ and $X$ proper, the degree map $\sum n_i [P_i] \mapsto \sum n_i [k(P_i) : k]$ is well defined on $CH_0(X)$.
Non-example: replacing rational equivalence by algebraic equivalence gives a different group.
For an elliptic curve $E$ over an algebraically closed field $k$, $CH_0(E) \cong \Z \oplus E(k)$, while modulo algebraic equivalence the $0$-cycles form only $\Z$, detected by the degree.
For a smooth variety, $CH^1(X)$ is isomorphic to the Picard group of $X$.