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  "body": "\n## Summary\n\nA motive is an object of a category built from algebraic varieties and [[algebraic cycles|9VFWPZEVGN]] on their products.\nGrothendieck proposed this category as a universal cohomology theory for varieties.\nEvery good cohomology theory, such as [[ℓ-adic cohomology|E1DVHN06CR]], [[algebraic de Rham cohomology|13F5JZPDD0]] or singular cohomology, should factor through the category of motives.\nPure motives over a field are well defined, with morphisms given by correspondences modulo an equivalence relation such as [[rational equivalence|K76N026JFK]].\nMany of their expected properties depend on the [[standard conjectures|D8GA2R3QT5]].\n\n## Explanations\n\n### L1 human\n\nDifferent cohomology theories of the same variety give groups that look alike, with the same dimensions and similar structures.\nGrothendieck asked for a single object behind all of them, the motive of the variety, of which each cohomology is a \"realization\".\nMotives can be split into pieces even when the variety cannot.\nFor example, the motive of the projective line is the sum of the motive of a point and the Lefschetz motive $\\mathbb{L}$.\nThe piece $\\mathbb{L}$ accounts for the cohomology in degree $2$.\n",
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