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  "body": "\n## Definition\n\nLet $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field and let $r \\ge 0$ be an integer.\nLet $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$ and $\\operatorname{Rat}_r(X)$ its subgroup of cycles [[rationally equivalent|K76N026JFK]] to $0$.\nThe *Chow group* of $r$-cycles on $X$ is the quotient\n$$CH_r(X) = Z_r(X) / \\operatorname{Rat}_r(X).$$\nIf $X$ is equidimensional of dimension $n$, the Chow group of codimension $j$ cycles is $CH^j(X) = CH_{n-j}(X)$.\n\n## Remarks\n\nExample: 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 $r$.\nA subvariety of dimension $r$ and degree $d$ has class $d$ times the generator.\nFor $r = 0$ and $X$ proper, the degree map $\\sum n_i [P_i] \\mapsto \\sum n_i [k(P_i) : k]$ is well defined on $CH_0(X)$.\nNon-example: replacing rational equivalence by algebraic equivalence gives a different group.\nFor 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.\nFor a smooth variety, $CH^1(X)$ is isomorphic to the Picard group of $X$.\n",
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