Chow group

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 $X$.