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  "body": "\n## Definition\n\nLet $k$ be a field, and let $\\mathcal{V}$ be the class of smooth [[projective varieties|A4H4CZ2V35]] over $k$.\nLet $X, Y \\in \\mathcal{V}$ with $X$ irreducible of dimension $d$, and let $r \\in \\Z$.\nThe group of *correspondences of degree $r$* is $\\operatorname{Corr}^r(X, Y) = CH^{d + r}(X \\times Y) \\otimes \\Q$,\nwhere $CH^\\bullet$ is the [[Chow group|EBEF6PEK4N]] graded by codimension, that is, [[algebraic cycles|9VFWPZEVGN]] modulo [[rational equivalence|K76N026JFK]].\nFor general $X$, sum over the irreducible components of $X$.\nThe composition of $f \\in \\operatorname{Corr}^r(X, Y)$ and $g \\in \\operatorname{Corr}^s(Y, Z)$ is\n$$g \\circ f = (p_{XZ})_*\\big(p_{XY}^* f \\cdot p_{YZ}^* g\\big) \\in \\operatorname{Corr}^{r+s}(X, Z),$$\nwhere the $p$ are the projections from $X \\times Y \\times Z$ and the product is the [[intersection product|4HV4WTAX9P]].\n\nThe category of *Chow motives* over $k$ with rational coefficients is defined as follows.\n\n1. Its objects, the *motives*, are triples $(X, p, m)$ with $X \\in \\mathcal{V}$, $p \\in \\operatorname{Corr}^0(X, X)$ such that $p \\circ p = p$, and $m \\in \\Z$.\n2. Its morphisms are $\\Hom\\big((X, p, m), (Y, q, n)\\big) = q \\circ \\operatorname{Corr}^{n - m}(X, Y) \\circ p$.\n3. Composition is the composition of correspondences.\n\nThe *motive of* $X$ is $h(X) = (X, \\Delta_X, 0)$, where $\\Delta_X$ is the class of the diagonal.\n\n## Remarks\n\nThe class $\\Delta_X$ is the identity of $\\operatorname{Corr}^0(X, X)$.\nThe *unit motive* is $\\mathbb{1} = h(\\operatorname{Spec} k)$ and the *Lefschetz motive* is $\\mathbb{L} = (\\operatorname{Spec} k, \\id, -1)$.\nExample: for $X = \\mathbb{P}^1$, let $e_0 = [\\mathrm{pt} \\times \\mathbb{P}^1]$ and $e_2 = [\\mathbb{P}^1 \\times \\mathrm{pt}]$ in $\\operatorname{Corr}^0(X, X) = CH^1(X \\times X)_\\Q$.\nThey are orthogonal idempotents, and $\\Delta_X = e_0 + e_2$ because the diagonal and $e_0 + e_2$ both have bidegree $(1, 1)$ in $CH^1(\\mathbb{P}^1 \\times \\mathbb{P}^1) \\cong \\Z^2$.\nThis gives $h(\\mathbb{P}^1) = (X, e_0, 0) \\oplus (X, e_2, 0) \\cong \\mathbb{1} \\oplus \\mathbb{L}$.\nNon-example: $(X, 2\\Delta_X, 0)$ is not a motive when $X$ is nonempty, since $2\\Delta_X \\circ 2\\Delta_X = 4\\Delta_X \\ne 2\\Delta_X$, so clause 1 fails.\nReplacing rational equivalence by homological or numerical equivalence gives other categories of pure motives.\n",
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