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  "body": "\n## Definition\n\nLet $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field, or more generally a scheme of finite type over a field.\nLet $r \\ge 0$ be an integer.\nAn *$r$-cycle* on $X$ is a finite formal sum $Z = \\sum_i n_i [V_i]$.\nHere the $n_i$ are integers and the $V_i$ are irreducible closed subvarieties of $X$ of [[dimension|36X2WTHYCB]] $r$.\nThe $r$-cycles form the [[free abelian group|0YMSEZE1VH]] $Z_r(X)$ on the set of such subvarieties.\nAn *algebraic cycle* is an element of $Z_*(X) = \\bigoplus_{r \\ge 0} Z_r(X)$.\nIf $X$ is equidimensional of dimension $n$, one also writes $Z^j(X) = Z_{n-j}(X)$ for the cycles of codimension $j$.\n\n## Remarks\n\nA cycle is *effective* if all $n_i \\ge 0$.\nLet $W \\subset X$ be a closed subscheme whose [[irreducible components|SY2MWC434Z]] $W_1, \\ldots, W_s$ all have dimension $r$.\nIts *fundamental cycle* is $[W] = \\sum_i m_i [W_i]$, where $m_i$ is the [[length|V6JBKT168T]] of the local ring of $W$ at the generic point of $W_i$.\nExample: in $X = \\mathbb{P}^2$, with $L$ a line and $C$ a conic, $2[L] - [C]$ is a $1$-cycle.\nThe subscheme defined by $x^2 = 0$ is the double line, with fundamental cycle $2[L]$ for $L = \\set{x = 0}$.\nNon-example: the union $L_1 \\cup L_2$ of two distinct lines is not one of the generators, since it is not irreducible; its fundamental cycle is $[L_1] + [L_2]$.\n",
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